GENETICS

Chance and the Distribution.
of Families.

In genetic experiments with plants and other animals, the most direct inferences as
to method of inheritance come from a count of the progeny. For example, if one'
were to make a cross betwee~ two pure lines. intercross the FJ,and from the inter-.
cross obtaip approximately three-fourths ofone parental type and one-fourth of the
other in thl! F z, a simple Mendelian model. one locus witbJ»'P...".aJJeJes...!'lnd-c1Qmi.".
,-llilllc.e-~oJJiJ1..b.elI1kII.~~ This conclusion would be reinforced if the backcross of
an Fr hybrid (to the pure-line parental type that had not appeared in the F 1) pro.:
duced progeny of which approximately one-half resembled the. aforementioned;
parental ty)X's and one-hctlf the Fr hybrid.
In man, the number of progeny from any single family is usually so ;m~ll that.
these conclusions would not be warranted. A mating of two .heterozygotes for a.
recessive defect can produce 4:0, 2:2, I :3, and 0:4 ratios, as ,well as the expected
J: I, in families offour children. Similarly, the mating A/a X a/a having two chil
dren, can produce 2:0 and 0:2 ratios as well as the expected 1:1.
A bit ofreflection will convince the reader that this is not surprising. Consider., .
for example, the couple with genotypes A/a and a/a, respectively, having two chil
dren. Genetic theory expects them to have one A/a child and one a/a child because
the A/a parent is expected to produce 1/2 A gemetes and 1/2 a gametes. ffthe A/a.
parent is the male, this ratio will usually be realized among the sperm. Is there,'
however, any guarantee that these"sperm will take turns, so to. speak, in fertilizing
the egg'! Obviously not. The sperm involved in producing the two children could.
well be A sperm in both cases or a sperm in both. If the A/a parent is the female, ,
there is not even a guarantee that the gametic ratio will be 1/2 A and 1/2 a, since !
each meiosis ordinarily produces only one gamete and the various meioses are
independent events: what happens in one meiosis does not influence what is to
happerl the next or any subsequent one. Hence, it could happen easily that the two'
children of this A/a X a/a mating are both A/a or both a/a.
Clearly the prediction of genetic results is fraught with uncertainties, events
over which no one has control. Such uncertainty is generally referred to as'
"chance." One author has recognized the large element of chance involved byenti.
t1ing his book <?n genetics The Dice ofDestiny. The reader may well ask: how can .
171 !
.t
I

f

;
S I
J
:

Chance and (he Distribution
bf Families ..


to method of inheritance come from a count of the progeny. For example, if one

were to make a cross between two pure lines, intercross the Flo :and from the inter-;

cross oblaip approximately three-fourths of one parental type a~d one-fourth of the,

other in lh. F 2• a simple Mendelian model. one locus with tw~ alleles and domi-I

nance. wouid be inferred. This conclusion would be reinforced if the backcross of·

an F,hybrid (to the pure-line parental type that had not appeared in the F,) pro- i

duced progeny of which approximately one-half resembled the aforementioned I

parental tyIXs and one-half the F, hybrid. : '

In man, the number of progeny from any single family is usually so small that..
these conclusions would not be warranted. A mating of two heterozygotes for a I
recessive defect can produce 4:0, 2:2, I: 3, and 0:4 ratios, as ..Jell as the expected :
3: I, in families of four children. Similarly, the mating A/a X a/a having two chil- I
dren, can produce 2:0 and 0:2 ratios as well as the expected I: Ii. i

A bit of reflection will convince the reader that this is not s'u·rprising. Consider, '
for example, the couple with genotypes A/a and a/a, respectively, having two chil
the A/a parent is expected to produce 1/2 A gemetesand 1/2 a gametes. If the A/a
parent is the male, this ratio will usually be realized among the sperm. Is there,
however, any guarantee that these sperm will take turns, so to speak, in fertilizing
the egg'! Obviously not. The.sperm involved in producing the two children could
well be A sperm in both c:;ases or a sperm in both. If the A/a parent is the female,
there is not even a guarantee that the gametic ratio will be 1/2 ~ and 1/2 a, since
independent events: what happens in one· meiosis does not influence what is to
happed the next or any subsequent one. Hence, it could happen basily that the two
Clearly the prediction of genetic results is fraught with uncertainties, events I
I
over which no one has control. Such uncertainty is generally referred to as
"chance." One author has recognized the large element.ofchance..involved byenti
tling his book on genetics The Dice ofDestiny. The reader may rell ask: how can
177

....-=----~ ~-

Ii

/

(a)

(e) (d)

(e) (f)

Figure 3.10 Examples of some simple Mendelian traits in hJmans that are relatively bommon: (a)common ba!dness, (b)
chin fissure, (c) ear pits, (d) Darwin tubercle, (e) congenital ptosis, (f)epicanthus, (g)camptodactyly, and (h) mid-digital
hair. See Table 3.1 for descriptions and modes of inheritance.,
I',

SIMPLE MENDELIAN INHERITANCE 11

c

Ii
,
f
Fig. 1-'4 A-C Nail defects in the nail-patella syndrome (anonycho-osteo-dysplasia). In
A they are most severe on the index fingers and (not shown) the thumbs. The little
fingers appear normal. B. Dystrophy of thumbnails in an affected brother. The lunulae
are abnormally large. C. Complete absence of thumbnails in a daughter of the patient
in A and in her eight-year old son. O-G. Some of the bone defects encountered in the
nail-patella syndrome. O. Absence of the patella in the eight-year old boy whose nail
defects are shown in C. (The epiphyseal centers of his femur and tibia have not yet fused
to the shafts.) The boy's mother has a patella (E) but suffers greater difficulties in walk
ing because the patella-and the associated extensor tendon-is displaced laterally. In
other affected the patella may be in normal position but hypoplastic. F and G: Typical
elbow defects. In F the head of the right radius. somewhat abnormally shaped, is dis
placed fdrward; in G the left radius, similarly abnormal, is displaced backward. In other
cases the capitulum of the humerus is poorly developed, compounding the problem.
Note also, in F, the exostosis of the coronoid process of the ulna. (A-D courtesy of L. S.
Wildervanck. E-G from Wildervanck, 1950b; courtesy of L. S. Wildervanck and Acta
Radioiogica. )

(a)

(e) (d)

(h)

~~~f
(e) (i) I
Figure 3.10 Examples of some simple Mendelian traits in humans that are relatively common: (a)common ba!dness, (b)
chin fissure, (c) ear pits, (d) Darwin tubercle, (e) congenital ptosis, (f)epicanthus. (g)camptodactyly, and (h) mid-digital
hair. See Table 3.1 for descriptions and modes of inheritance.'

SIMPLE MENDELIAN INHERITANCE 11

Fig. 1-'4 A-C Nail defects in the nail-patella syndrome (anonycho-osteo-dysplasia). In
A they are most severe on the index fingers and (not shown) the thumbs. The little
fingers appear normal. B. Dystrophy of thumbnails in an affected brother. The lunulae
are abnormally large. C. Complete absence of thumbnails in a daughter of the patient
in A and in her eight-year old son. D-G. Some of the bone defects encountered in the
nail-patella syndrome. D. Absence of the patella in the eight-year old boy whose nail
defects are shown in C. (The epiphyseal centers of his femur and tibia have not yet fused
to the shafts.) The boy's mother has a patella (E) but suffers greater difficulties in walk
ing because the patella-and the associated extensor tendon-is displaced laterally. In
other affected the patella may be in normal position but hypoplastic. F and G: Typical
elbow defects. In F the head of the right radius, somewhat abnormally shaped, is dis
placed forward; in G the left radius, similarly abnormal, is displaced backward. In other
cases the capitulum of the humerus is poorly developed, compounding the problem.
Note also, in F, the exostosis of the coronoid process of the ulna. (A-D courtesy of L. S.
Wildervanck. E-G from Wildervanck, 1950b; courtesy of L. S. Wildervanck and Acta
Radiologica. )

1-6. Anonycho-osteo-dysplasia, better known as the nail-pateJla syndrome (Fig. 1
4), is one of the traits listed in Table 1-3. <!:all it the np locus. A man and his sister
both have the trait, and both marry persons who lack it. The man's son, who lacks
it, marries the sister's daughter, his cousin( who does have it. (a) What is the out
look for their children? (b) If their first ctlild is normal with respect to this trait,
what is the outlook for their next child?
1-7. Suppose you examine the widowed mother of the man add his sister of exer
cise 1-6 and you find that she lacks the nail-patella syndrome. What would you
conclude about the phenotype and genotype of her late husband (their father)? If
more than one answer is possible, which islthe more likely, and why?

Table 1-3 Progeny from testcrosses for several dominant traits (For references see
Table 5-1 of Levitan and Montagu, 1977) ,
Num~r of Test
Trait Cross Sibships Normal Affected Total
Anonychia with ectrodactyly (Fig. 1-9) 139 57 66 123
ElliptocytosIS (ovalocytosis). both loci '65 113 99 212
(Fig. 13-2)
Epidermolysis bullosa, all dominant forms 50 80 62 142
i
(Fig. 1-3)
Nail-pate~la syndrome (Fig. 1-4) 157 268 288 556
TOlal 31 518 515 1033
Ratio 1.006 :

3.29. Infantile amaurotic idiocy (Tay-Sachs disease) is a recessive hereditary abnormality causing death within
the first few years of life only when homozygous (ii), The dominant condition at this locus produces a
normal phenotype (/-), Abnormally shortened fingers (brachyphalangy) is thought to bidue't~- a 'genoi"ype
heterozygous for a lethal gene (BB L ), the hom~~ygote (Sii) being~riormal, and the-9th~rfi(;;;;~Jyg9.!~JBLBL)
be~r:tgJetha.L What are the phenotypic expectations among teenage children from parents_who ,are both
brachyphalangic and heterozygous(or''nfantileamaurotic idiocy?
. '-~' '.~ ..--"" '-"--_'_- ..
- , , .

3.30. In addition to the gene governing mfantile amaurotic idiocy in the above problem, the recessive genotype
of another locus (jj ) results in death before age 18 due to a condition called "juvenile amaurotic idiocy."
Only individuals of genotype /-J- will survive to adulthood, (a) What proportion of the children from parents
of genotype liJj would probably nol survive to adulthood? (b) What proportion of the adult survivors in
part (a) would not be carriers of either hereditary abnormality?

.. "

Table 1-3 Progeny (rom testcrosses (~r several domina,nt tr~its (For refer~nces see
Table 5-1 of Levitan and Montagu, 1977) i
Number of Test
Cross Sibships Normal ' Affected Total
Trait
65 113 99 212
. Elliptocytosis (ovalocytosis), both loci .j
~ r 1
(Fig. 13-2)
50 . ~ 80 62 142
Epidermolysis buJlosa, all dominant forms
,
(Fig. 1-3)
157 268 288 556
Nail.patella syndrome (Fig. 1-4)
311 518 515 I 033
Total
Ratio 1.006 :

Table 1-4 Offspring in 416 marriages b¢tween various M·N blood types. Data (rom
Wiener et aJ. (1963).
Progeny Total Number of
Father Mother M M:N N ! • Offspring Families

M M 71 1" 0 72 42
N N 0 0 29 29 20
M N 0 43' 0 43 23
N M 0 24 0 24 i3
All M X N 0 ~
67 , 0 67 36
M MN 67 46 0 113 63
MN M 60 55 0 115 59
All M X MN 127 IOIj 0 228 122
N MN 0 31 44 75 39
MN N 0 40, 27 35
All N X MN 0 71) 71 142 74
MN MN 61 118 53 '232 < 122
Totals 259 358 ' 153 770 416
, ,
I
3This apparent contradiction to the laws of heredity is believed to be owing t~ ,illegitimacy. but it may represent
a new mutation, a change in the genetic material of one of the parents. '

--.~-
--~~ .. .~.

./it

1-6. Anonycho-osteo-dysplasia, better known as the nail-patella syndrome (Fig. 1
4), is one of the traits listed in Table 1-3. Call it the np locus. A man and his sister
1 • '
both have the trait, and both marry persons who'lack it. The'man's son, who lacks
it, marries the sister's daughter, his cousin, who does have it. (a) What ,is the out
look for their children? (b) If their first ehild is normal witt{ respect to this trait,
what is the outlook for their next child? '
1-7. Suppose y6u examine the widowed ,mother of the man and his sister of exer
cise 1-6 and you find that she lacks the nail-patella syndrome. What would you
conclude about the phenotype and genotype of her late husband (their father)? If
, 1
more than one answer is possible, which is the more likely, and why?

Table 1-3 Progeny from testcrosses for several dominant traits (F'or references see
Table 5-1 of Levitan and Montagu, 1977) .
i'

Number of Test !
Trait CrosslSibships Normal Affected Total

ElliptocytosIS (ovalocytosis). both loci i 65 113 99 212
(Fig. 13-2)
Epidermolysis bullosa, all dominant forms 50 80 62 142
(Fig. 1-3)
Nail-patella syndrome (Fig. 1-4) 157 268 288 556
Total 311 518 515 1033
Ratio 1.006 :

3.29. Infantile amaurotic idiocy (Tay-Sachs disease) is a recessive hereditary abnormality causing death within
the first few years of life only when homozygous (ii). The dominant condition at this locus produces a
normal phenotype (/-). Abnormally shortened fingers (brachyphalangy) is thought to be due to a genotype
heteroz.ygous for a lethal gene (881..), the homozygote (88) being normal. and the other homozygote (8 L8 L )
being lethal. What are the phenotypic expectations among teenage children from parents who are both
brachyphaJangic and heterozygous for infantile amaurotic idiocy?

3.30. In addition to the gene governing infantile amaurotic idiocy in the above problem. the recessive genotype
of another locus (jj ) results in death before age 18 due to a condition called "juvenile amaurotic idiocy."
Only individuals of genotype /.). will survive to adulthood. (u) What proportion of the children from parents
of genotype IUj would probably not survive to adulthood'! (b) What proportion of the adult survivors in
part (a) would not be carriers of either hereditary abnormality? .

Table 1-3 Progeny from testcrosses for several domin~nt traits (For'references see
Table 5-1 of Levitan and Montagu, 1977)
Number of Test
Trait Cross Sibships t Normal Affected Total

Elliptocytosis (ovalocytosis), both loci 65 113 99 212
(Fig. 13-2)
Epidermolysis bullosa. all dominant forms,I 50 80 62 142
(Fig. 1-3) ,
Nail~patella syndrome (Fig. 1-4) 157 268 288 556
311 518 515 I 033
Total
Ratio 1.006 :

Table 1-4 Offspring in 416 marriages between various M-N blood types. Data from
, I
Wiener et al. (1963).
. Progeny lotal Number of
Father Mother M MN N Offspring Families
M M 71 I" 0 72 42
N N 0 0 29 29 20
M N 0 43 0 43 23
N M ~ 24 0 24 13 .-4

All M X N 0 67 0 67 36
M MN 67 46 0 113 63
MN M 60 5$
-, ~ 115 59
All M X MN 127 101' 0 228 122
..~.
N MN·.t 144 75 39
N 0 4gt 67
MNt ~ 35
All N X MN 0 71 71 142 74
MN MN 61 118 53 232 122
Totals 259 .358' 153 770 416
"This apparent contradiction to the laws of heredity is believed to be owing to illegitimacy, but it may represent
. a new mutation, a change in the genetic material of one of the parents. .

,
Table 8-1 Expected progeny in a random sample of four-children families vhere one
is A/a and the other a/a, locus A being autosomal
Expected Progeny
As Proportions
of Next
In Numbers~ Generation
Kind of Number of
Family ProbabililY~ FamiJies b A/a a/a A/a a/a

4 A/a 1/16 100 400 1/16
3 A/a, 1 a/a 4/16 400 1200 400 3/16 1/16
2 A/a, 2 a/a 6/16 600 1200 1200 3/16 3/16
I A/a, 3 a/a 4/16 400 400 1200 1/16 3/16
4 a/ans ---'1.L.§ 100 400 !L!..Q
Tolal 16/16 1600 3200 3200 8/16 8/16
1/2 1/2
'Calculated in the text.

bCalculated on the basis of 1600 families.

Table 8-2 Expected progeny in a random sample of four-children families where the
parents are heterozygous for a recessive gene a
Expected Progeny
As Proportions of
In Numbersb Next Generation
Kind of Number of
Family Probability' Families b A/-c a/a A/-< a/a

4,1/81/256 2025 8100 81/256
3.1/-,1 a/a 108/256 2700 8100 2700 81/256 27/256
2,1/-,2 a/a 54/256 1375 2700 2700 27/256 27/256
I ,1/-,3 a/a 12/256 300 300 900 3/256 9/256
4 a/a 1/256 25 100 1/256
Total 256/256 6400 19200 6400 192/256 64/256
3/4 1/4

'Calculated in the text.

bCalculated on the basis of 6400 families.

<.4/- means that the genotype is either A/d or A/a.

Table 8-3 Expected progeny in truncate sample of four-children families of which
the parents are both heterozygous for a recessive gene a
Progeny in the
Kind of Proportions of all Proportions of the Observed Sample
Family A/a X A/a Families' Observed Families A/ a/a
3.4/-, I a/a 108/256 108/175 81/175 27/175
2A/-,2a/a 54/256 54/175 27/175 27/175
I A/a, 3 a/a 12/256 12/175 3/175 9/175
4 a/a 1/256 1/175 ~
Total 175/256 175/175 111/175 64/175
63.4% 36.6%
·See Table 8-2.

r

r
/
Table 8-4 Analysis of the data in Table 8-3 by the direct $ib method (Weinberg's
general proband method) .
Total Sibs of Corrected
Expected "Observed" Sibs/Proband Probands Progeny
Affected Proportion" Proportion Probands Affected: Normal Affected: Normal Affected: Normal
0 81 0 0 0:0 0:0 0:0
I 108 108 I 0:3 0:3 0:324
2 54 54 2 1:2 2:4 108:216
3 12 12 3 2: I 6:3 72;36
4 4 3:0 12:0 • 12:0
Total 192: 576
Ratio 1:3
'Per 256 four-children families of A/a x A/a marriages in the population.

.,

Table 8-5 Suggested calculation of the~ gib Fl'IcrilOct:"'1l1ustration is for a
theoretical truncate distribution of four-child families from ~/a X A/a

Number of Number of Number of Corrected Progeny
Family Unaffected/Family Affected/Family Families Normal Affected
Type (U) (A) (N) UAN A(A - I)N
3: I 3 I 108 324 0
2:2 2 2 54 216 108
1:3 I 3 12 36 72
0:4 0 4 I 12
Total 576 192

Table 8-6 Direct sib correction in six families of spongy type polycystic kidneys of
onset. (Data from Lundin and Olow, 1 )
Unaffected Affected
Family Per Family Per Family Number of
Type (U) (A) Familie.s (N) UAN 'A(A - I)N
A I I 3 3 0
B 2 2 2 8 4
C 2 l ' I 2
Totals 8 5 6 21 6
Ratio 0.778 0.222
ExPected ratio 0.75 0.25
Expected numbers 20.25 6.75
x 2 : . 0.106
P;> 0.70

~8
·.~ .. - ':---.=.-.~

Chance and the Distribution
of Families

to method of inheritance come from a count of the progeny. For example. if one
were to make a cross between two pure lines, intercross the F" and from the inter
cross obtain approximately three-fourths ofone parental type and one-fourth of the
other in the F2• a simple Mendelian model, one locus with two alleles and domi
nance, would be inferred. This conclusion would be reinforced if the backcross of
an F, hybrid (to the pure-line parental type that had not appeared in the F,) pro~
duced progeny of which approximately one-half resembled the aforementioned
parental types and one-half the F, hybrid.
In man, the number of progeny from any single family is usually so small that
these conclusions would not be warranted. A mating of two heterozygotes for a
rece!>sive defect can produce 4:0, 2:2, I :3, and 0:4 ratios, as well as the expected
3: I, in families offour children. Similarly, the mating A/a X a/a having two chil
dren, can produce 2:0 and 0:2 ratios as well as the expected I: I.
A bit ofreflection will convince the reader that this is not surprising. Consider,
for example, the couple with genotypes A/a and a/a. respectively, having two chil
,

. _~,Il
the A/a parent is expected to produce 1/2 A gemetes and 1/2 a gametes. If the A/a
however, any guarantee that these sperm will take turns, so to speak, in fertilizing
the egg? Obviously not. The sperm involved in producing the two children could
well be A sperm in both cases or a sperm in both. If the A/a parent is the female,
there is not even a guarantee that the gametic ratio will be 1/2 A and 1/2 a, since
happen the next or any subsequent one. Hence. it could happen easily that the two
Oearly the prediction of genetic results is fraught with uncertainties. events
"chance." One author has recognized the large element ofchance involved by enti
tling his book on genetics The Dice ofDestiny. The reader may well ask: how can
~ 177
~.
-elY.i

/ ',.
178
'.
c TEXTBOOK OF HUMAN GENETICS CHANCE AND THE DISTRIBUTION OF FAMILlES 179
genetics pretend to be a science if it cannot really predict the results of a given failure. The chance of tossing a coin heads, for example, is I: I. one chance of suc
mating? Or. to put it more concretely, since chance plays such a large part in genetic cess to one chance of failure; similarly the odds of obtaining a five in the toss of a
transmission. of what value are the stated Mendelian ratios? The question becomes die are I :5. one chanee of success to five chances of failure. For arithmetic
especially pertinent. we shall discover below, when the family size becomes greater
ulation, however, the best way to state the probability is the way it is often instinc
than two: 1110.1'1 families then will /lol yield the stated ratios of offspring.
tively given for the coin example: a~~aJ.1iQt1,..¥"hQs~dc:,Mminator is the nl!.T ber
Before answering this question it is necessary to understand that genetics in
...oLeg.l1,alLy_likely-e..f.ents-flossible-in-th@-osi-tuat,ion,..llJld,er discussion and whosc
being beset with uncertainties is not singular among the sciences. Every scientist
_n!l.menl1o~is..the.JlUll1,q~J_qlt.he~e_that~Qnsli!uJe the event whoscfik"ClTil'O'OO is in
knows that in the real world absolute certainty does not exist: there arc only relative _q_ue~tion.,.
degrees ofuncertainty. To pick a rather absurd example, one could ask the follow.
Example: Suppose an urn contains five balls of identical size and shape. Two
ing two questions of a meteorologist: (I) will the sun rise tomorrow'? and (2) will
arc black, marked B I and B2, and three arc white, marked WI, W2, and WJ.
tomorrow be a sunny day? He cannot answer either question with absolute ccr.
respectively. We reach in to take out one ball. (I) What is the probability of obtain
but it is clear that he can answer the first with much more confidence than ing the ball marked W2; and (2) what is the probability of obtaining a white ball')
the sl'cond.
A great stride in the progress of a science is mad~: when it can ascertain or "I' I
measure these relative degrees of uncertainty. The mc:eorologist feels a sense of ( I) P ro b ab 1 Jty 0 f W2 = ___
n_um_be_r_o_f_W_2_b_al_ls__
total number of balls present 5
achievement when he can improve his forecast from "it may rain" or even '"it will
probahly rain" to "there is a 90 percent chance ofrain:'... JJ'cpmeasurement or quan.
1 number of white balls 3
ti.l;lti.Ql}"Qf~uncc:natnt~iS"eallt'a..f~t:61t)alJiJUJ;" (2) Probability of white = total number of balls present -
5
,._' " Usually the positive aspect of probability is emphasized. We arc more inter.
ested in the degree we have moved toward certainty than in the amount of uncer Mathematically. these statements would be written:
tainty left. Hence we think of probability as the likclihoGd ofa desired event rather
than as the degree to which we fall short of certainty in attaining it.
P (W2) = 1/5:
Let us examine some of the basic properties of probability. Perhaps we should
P (white) = 3/5.
hegin with a well-known example. If a coin were to be tossed and we were asked P represents the phrase "probability of' the item in parentheses.
the probability of its turning up "heads" the immediate answer would be some •. How large or now small may probabilities be? Thc....maximum probability
'ariant of "tinY-fifty:' "one to one" or "one-hal(" If we asked why we gave this would be reached if we were satisfied with obtaining any ball present. wfictliCr"'ii
answer the reply would be roughly: why. it stands to reason; there are two ways the was white or black. Thus,
coin may fall and one of these is heads, so the chance is I out of 2. one-halfor one
10 one. If we had an evenly weighted die and wished to know the probability of its P (white or black) = 5/5 = I.
upright face showing five dots when thrown, it does not seem difficult to extend the
logic of this answer: since any of the six sides of the cube could turn up and the Similarly, the mlfllmum probability would come from wishing to obtain so 111 t'
l:lCe with five dots is only one of these, there is one chance out of six of obtaining other color than white or black from this urn:
the "Ii'e." Clearly. the proba 1 ility of success, that is. the likelihood that a desired
l'Cnt will happen, depends on the number of alternative events that could happen /' (neither white nor black) = 0/5 = O.
and on the number of these that spell "success."
This indicates that probabilit~ is a positive number between 0 and I.
Merely saying "the number of alternative events" C:.tll be misleading. however.
The sophisticated reader will suggest that probabilities of I and 0 repn:sl'nt
heca lISC one could argue that in throwing the die on a gi ven throw there arc only
"certainty," certainty of success and certainty of failure. rl'spectively, This is cor
two alternatives. success or failure, obtaining a five or not obtaining a five. and this
rect. Certainty is possible in a mathematical sense, since a mathematical system
lead to the belief that the probability of SUCcess was one of two. Obviously
assumes that only the conditions postulated will occur. In a few simple real situa
this is not true: for every chance of obtaining a five there arc five chances of obtain
tions this degree of simplicity may be approximated-but there is never a guaran
il'g something else. We must modify our statement. therefore, to state that the
tee that the postulated condition will prevail. Thus we may feci secure that there
lllJ.l.bah.il~t..j;.,Qf success depends on the number of eqll(/I/J;,.lwn/Jahil' (or equally likely.
are two black and three white balls in the urn, and this usually docs approximall'
:1~1b,(ll1alicians-sa.Y;)..Go;€'nts"'fl0ssihlG-and_Lh~!lumber of these""'1IUi'l'Spell
~~l;CS,"- _ a mathematical system. but we really cannot guarantee absolutely, for example.
that one or another of the balls did not disintegrate after wc placed it in the urn. III
The probabilities discussed above may be stated "one chance out of two" and
practice, tl},elikelihoQd.of.su~c,~af,l~ull~lI,.pec.te_d.e,y,glltjs_oO,cn so small that we ignore
" lilt'cham'!' out of six." They could also be phrased as relative odds of success and
it and we do speak of probabilities of I or 0 even for living things. Thus we have

"
"-"""J.-______

8
,,! .:.

'

0 "
-

a
M
a Chance andthe Distribution
CT of Families
o

"

to meJhQd, of inheri!anfe~come from a count of tQ~_l'-rogeJ:lY. For example;-if one
were to make a cross between t"'QPureljJle.s, inten::ross tl!e. Fh and from the inter
cross obtain approximately three-fourths ofone parental type and one-fourth ofthe
other in the F2, a simple Mendelian m~del, onc;Jp~l!s with two allel~~ ;~d do~i
nance, would be interred. This conclusion would be reinforce<rifthe backcross of
an-F,hybrid (to the pure-line parental type~thai had-not appeared in ti!e F,) pro
ducedpr()geny~f which appro~imately ol,le-6a-lf resembled ·the' a'forementioned
parental types and one-half the F, hybri4.
In man, the number of progeny from any single family is usually so small that
these conclusions woui<f noi'be" ~arr;.~ted Amating-of t~o heterozygotes for a
~<:e.~si~e_d~fectca'!.PIo_<!uce 4:0,2:2, 1:3. and 0:4 ratio.s. as wella's the expected
.3: I., in families of four children. SimilarlY. the mating A/a X a/a having two chil
dren, 'can produce '2:0 and 0:2 ratios as well-as the' expected I : I. . ". "
. - . A bit of reflection wiUoonvince the reader that this is not surprising. Consider,
for example, the couple with genotypes A/a and a/a, respectively. having two chil
the A/a parent is expected to produce 1/2 A gemetes and 1/2 a gametes. If the A/a
however, any guarantee that these sperm will take turns. so to speak, in fertilizing
the egg? Obviously not. The sperm involved in producing the two children could
well be A sperm in both cases or a sperm in both. If the A/a parent is the female,
there is not even a guarantee that the gametic ratio will be 1/2 A and 1/2 a, since
happen the next or any subsequent one. Hence, it could happen easily that the two
children ofthis A/a X a/a mating are both A/a or both a/a.
Clearly the prediction of genetic results is fraught with uncertainties, events
"chance." One author has recognized the large element ofchance involved by enti
tling his book on genetics The Dice of Destiny. The reader may well ask: how can
117
:

;> ". '&
178
TEXTBOOK OF HUMAN GENETICS CHANCE AND THE DISTRIBUTION OF FAMILIES 179
g('netics pretend to be a science if it cannot really predict the results of a given
failure. The chance of tossing a coin heads, for exam pic, is I: I, one chance of suc
mating'.' Or, to put it more concretely, since chance plays such a large part in genetic
cess 'to one chance of failure; similarly the odds of obtaining a fivc in the toss of a
transmission. of what value arc the stated Mendelian ratios? The question becomes
die are I :5, one chance of success to five chances of failurc. For arithmetic manip
cspecially pertinent. we shall discover below. when the family size becomes greater
ulation. however. the best way to statc the probability is the way it is oftcn instinc
than two: most families then will flot yield the stated ratios of offspring.
tively given for the coin example: as a .fractioQ~,ho~cIdC'iiQiiilnatodi!£_lbe_number
Aefore answering this question it is necessary to understand that genetics in
QLtqlJ.alh: likcJLe~yenJs_possjblc_in Jhc~iJ.!.Iali.OT'l_!.I.!1der discussion and who..:;e
hcing beset with uncertainties is not singular among the sciences. Every scientist
in umerato~ is the n umber oLtl;le~e_thill, cQ1Jstitllle.tl1e.eveIlLwhosdi kcl.i hoodj!!jn
knows that in the real world absolute certainty docs not exist: there arc only relative question,.
dcgrees of uncertainty. To pick a rather absurd example, one could ask the follow-
Example: Suppose an urn contains five balls of identical size and shape. Two
two questions of a meteorologist: (I) will the sun rise tomorrow'! and (2) will
arc black. marked 81 and 82, and thrcc arc white. marked WI, W2, and W1.
tomorrow be a sunny day? He cannot answcr either question with absolute cer
respectively. We reach in to take out one ball. (I) What is the probability of obtain
tainty. but it is clear that he can answer the first with much more confidence than
the sccond . ing the ball marked W2; and (2) what is the probability of obtaining a white ball')
.. great stride in the progress of a science is mad~: when it can ascertain or number ofW2 balls ~
measure thesc relative degrees of uncertainty. The mc:eorologist feels a sense of (1) Probability of W2 = total number of balls present 5
achicvement when he can improve his forecast from "it may rain" or even "it will
probably rain" to "there is a 90 percent chance of rain. " The measurement or quan numbc·r of white balls ~
titation of uncertainty is called !1 rohahility. (2) Probability of white = total number of balls present 5
Usually the positive aspect of probability is emphasized. We arc more inter
ested in the degrec we have moved toward certainty than in the amount ofuncer Mathematically. these statements would be written:
tainty leti. Hence we think of probability as the likelihood ofa desired event rather
than as the degree to which we fall short of certainty in attaining it. /,(W2) = 1/5;
P (white) 3/5.
Let us examine some of the basic properties of probability. Perhaps we should
hegin with a well-known example. If a coin were to be tossed and we were asked
@,=eprescJ)ts.th~ph(a~g"::'lrQbl'lJ:tililJ'...QC the item in pnrentheses.
the probability of its turning up "heads" the immediate answer would be some
How Inrge or how small may probabilities be? The maximum probability
ariant of "fiftY-fifty," "one to one" or "one-half." If wc asked why we gave this
would be reached if we were satisfied with obtaining any ball present, whether it
answer the reply would be roughly: why. it stands to reason: there arc two ways the was white or black. Thus,
coj n may fall and one of these is heads. so the chance is lout of 2, one-half or one
to one. If'e had an evenly weighted die and wished to know the probability of its P (white or black) = 5/5 =<i)
upright fnce shOwing five dots when thrown. it docs not seem difficult to extend the
logic of this answer: since any of the six sides of the cube could turn up and the Similarly, thc rninimuQlJll·Q.babiJity would come from wishing to obtain some
nll'c with fie dots is only one of these. there is one chance out of six of obtaining other color than white or black from this urn:
the "five." Clearly. the proba 1 ility of success, that is. the likelihood that a desired
l'Cnt vill happen. depends on the number ofalternative events that could happen P (ncither white nor black) = 0/5 =(2)
and on the number of these that spell "success."
This indicates that J~'=9bability is a posi!ive number betweeo!b and I)
Tvh.'rdy saying "the number of alternative events" can be misleading. however.
The sophisticated readcr will suggest that probabilities of 1 and 0 represent
because one could argue that in throwing the die on a given throw there arc only
"certainty," certainty of success and certainty of failure, respectively. This is cor
two a!ternati·es. success or failure. obtaining a five or not obtaining a five. and this
rect. Certainty is possible in a mathematical sense. since a mathematical systelll
might lead to the belief that the probability of success was one of two. Obviously
assumes that only thc conditions postulated will occur. In a few simple real situa
this is not true: for cvery chance of obtaining a five there arc five chances of obtain
tions this degree of simplicity may be approximated-but there is never a gUllrnn·
II'g something else. We must modify our statement, therefore, to state that thc
tec that thc postulated condition will prevail. Thus we may feci secure that there
pronability ofSuccess depends on the number of ('quail)' /lrobah/c (or equally likely.
arc two black and thrce white balls in the urn, and this usually does approximate
as thc mnthelllaticians say) events possible and the number of thesc that spell
SUCt'l'ss. a mathematical system, but we really cannot guarantee absolutely. for example,
that onc or another of the balls did not disintegrate after we placed it in the urn. In
The probabilities discussed above may be stated "one chance out of two" and
chance out of six." They could nlso be phrased as relative odds of success and
"Illlt'
practice, the likelihood of such an unexpected event is often so smalUl}a,t weigno!:r
it. and we do spcak of probabilities of i or 0 even for living things. Thus we h:lVl'

. ~
,.~
l1:fO TEXTBOOK OF HUMAN GENETICS CHANCE AND THE DISTRIBUTION OF FAMILIES 18

seen already that we,ignore the p-robabililY of new mutations when we predict that Urn A Urn B Urn 1 Urn B Urn A Urn [j Urn 1 Urn R Urn 1 Urn B
the cross: p
0)G) CDG) 0)G) (0G) QG)
A/A X :1/:1 - all A/A. 0)0 (00 0) 0' (00 80
In probability terms we arc saying
G)G) CDG) 8G) (0G) 0G)
P (A/A X ..1/.·1 - A/A) = I.
and this cross is therefore usually wntten
0)0 (00 80 (00 00
A/A X A/A - I A/A.
Q. Q CDQ 8Q 0Q QQ
Implicit in asking the probability of obtaining a Il-hill' ball from the urn is the 88 08 8 8 08 08
Idea that il docs not matter whether we obtain W I. Wi. or W3. Any of these alter
n:! I i Vl' e'en Is spells success. Thus. Fig. 8-1 The 30 combinations possible when a ball is drawn from each of two urns i
the first urn contains 2 black balls and 3 white balls, the second contains 3 black Oil!'
P (white) ==
I + I + I and 3 white ones, and the balls are equal in size but distinguishable by numbpr.

1/5 + 1/5 + 1/5.
Thus.

Hut.
x3

P(WI)= 1/5.
P (black. black) = 5 X 6
I' (W2) = 1/5. and
.; 2 3
P (W3) = 1/5.
,= 5 X'f;'
Iknec.
But.

P(white) = peW!) + P(W2) + P(W3).
'2

111 ot her ords.jla II erelll has S(xccqLalt~p.wtiJ;G,J'1J;w.);J!oJla.Uaj,n m.c.n.t..,Q,f a llt. ~_~ = I' (black from urn A)
a Ill'rna I i e is considered at tai n,1l1cn.l~ofTt.h('TdesiFC&eVcrr1'!'"tht?"flf(-)/!ahilil,h.4).t,;.IlC,1:~.~
i~1'i11'7f1TIi(' [I}'{)bahiliI ic..!i-!J£Jjw.(!_lJi/'CI;lJati-w.l•.J(J!~!1IS-J n
com monsen.'ie terms, and
when we arc nol particular or choQsy the chances for success areincreased. 'J
An equally interesting probability is that of a success which is composed or P (black from urn B).
6
scwral eents thaI must happen simultancously or ill succession. Clearly, this
prohahility must be smaller than the likelihood of attaining anyone of the events Thus. the probability of obtaining black balls from both urns is the product ()
alone. How much smaller? To solve this problem let us relurn to the urn with the the probabilities or obtaining a black baH from each urn.
10 hlack halls and the three white ones and set up a second urn with three black We may generalize this into a rule:.J1l£'prohahilil.1' Oro/Jlaillillg ('('rtaill sill/III
halls. 83. 84. B5, and Ihree white ones. W4, WS. and W6. If a ball is to be removed f a 11(:tJ.IIJ-(Qt-s,ucc~si~c:,I:J:l11..W.l--llJ.(;'.J1lJ.1d.IJ5;t....Q[ 11U:"'IID1!i'tilri1TTif'n'JiTl/(· (w·n I.'
from each urn. how likely is it. for example. that both will be black? To answer this illl·()lJ.;(:lLJ.J&..chcck,Jbi.s~generalization. we may note that.
c may return to our basic concept of probability and enumerate how many -
l'qually probable events. each consisting of one bal! from urn A and one ball rrom I ) (white. w h'
. Itc) = '3 X (;
5 .1
urn B. can happen and no Ie how many of these consist ()f a black ball from each
urn. The equally probable events arc shown in Fig. 8-1. 9
"" 30
There arc .10 equally probable events and 6 of these represent pulls of two black
1.,111 .... TIm'>,
and this corresponds to the enumerated proportion.
P (black. black) ;", 6/.10 = 1/5. . This rule too. fits our commonsense expectations. The likelihood that both wil
happen must be smaller than the likelihood ofcither one alone. I'or each c )uJd Oftl'l
NOlI.' Ihal Ihe 30 in the denominator is the product of the number or balls in urn
happen without the other. Since probabilities are fractions less than one, multipl~
-. : and the number of balls in urn B: likewise, the 6 in Ihe numerator is the product
ing them has the effect of producing a still smaller fraction. tha,1 is. decreasing th,
oflhe Ilumherofblack balls in urn A and the number of blacks in urn B.
no ' probabilitv.

•

Jil2
#
.J

.' This "'plUltipJkatioll~r,uk;;:";s valid whether the desired events occur simulta
ncously or in a particular order. Furthermore, it is applicable whether the proba
WHA T THE GENETIC RATIOS MEAN

hilitics of successive events arc independent or dependent. Drawing balls from the We may now consider the meaning of the Mendelian ratios.

tO urns-described above-is an example of independent probabilities. where the
rrohability of one event does not influence the probabilty of another in the series.
1I1ealling /: A genetic ratio sta(~.Llu:..pt;abaililil,l,-:lh[q~si!Jglc:.hil:!Jl. When we state

~ .
A/a X a/a ...... 1/2 A/a, 1/2 a/a
Likewise. if only one urn is used and the ball replaced between pulls. the probabil
ities of successive choices are independent. To illustrate tJ!:pendeflt nro.babililies we., we are saying that at any given birth from these parents the odds arc I: I that the
may consider only the first urn and calculate, for example. !!l$..n(ob@jLit¥~oipick child will be a/a. On the other hand, for the mating
in a black ball twice iIl..-succ~ssi0n..if-t·tu;:uirst~Q.Q.e..r~mo.xedjs not seen and 110t
replaced )efore m!.Uil'l~Q.tJt...lJ1e_sf.CQn.d_ ) Ala X A/a -+ 1/4 A/A. 1/2 II/a, 1/4 a/a.
"';;;';"Forrt~ choice. as before.
the stated ratio means that at any given birth there is twice as much chance that
P (l31 or 82) = 2/5. the baby will be A/a than that it will be A/A. Likewise the odds are 2: I for A/a
"""""'---"!""--=:;.:00Ii!'!"- versus a/a and I: I for A/A versus a/a. If A is dominant over a, the probability is
I f the first choice is successful. only four balls remain in the urn and only one of
3/4 that a child from the last mating will show the A phenotype. 1/4 that it will
I hese is black. Therefore. for the second choice
show a phenotype, in other words, three times as much chance for gene A to he
P (black) = [1.4......· present at least once than for its complete absence, a/a.
.:-.lI(l$).

Meaning 2:,..!.1kall,ll.,sJbsllip..+hfM'e,5uit..-suggested.lJ,v..Jl,u;...ge/l£,tic ratio has the W!.ldal
The probability that both events happen is. then
prohabili1.I. This probability is usually /lot the probability in the Mendelian ratio.
2/5 X 1/4 = 2/20 = 1/10 "-but must be calculated for each case.
The genetic ratio leads us to expect that the distribution of children in a family
The student is often disturbed by dependent probabilities because it seems
will conform to this ratio. but we know it usually will not do so. For example. if
more necessary than for independent probabilities to suppose that the first choice
the parents in the mating
is slIccessful before calculating the probability of the second. He should keep upper
most in mind that we arc seeking the likelihood that both events do happen. We A/a X a/a -> 1/2 :I/a. 1/2 a/a
wish to...calculate the probability of success. not the probability of failure. Failure.
not obi5ininii'he desired resuft"tndudes all the other results that could occur: the decide to have two children the ratio "predicts" that they will have one A/a child
lirst choice is "right" but the second "wrong." the first wrong even if the second is and one a/a child. But our biological knowledge, and observation of actual fami
right. or both choices are wrong. Calculating the probability of failure on the same lies. tells'us that such families will consists of two A/a children or two a/a children
basis used for the probability of success verifies our result: as well as one A/a and one a/a. Using our rules of probability and the first meaning
for the genetic ratio, we can calculate the probability or expected relative frequency
I' (black. white) = 2/5 X 3/4 = 6/20 of these sibships.
P (white. black) 3/5 X 2/4 = 6/20
P (white. white) = 3/5 X 2/4 = 6/20 , 1'(2:1/a) = /'(/I/a. A/a) = 1/2 X 1/2 = 1/4.
Sinn' any of these alternative spells failure. Likewise.
P (Failure) = 6/20 + + = 18/20.
Since
6/20 6/20
)Jjgj£) = P(a/a. a/.£l;. 1&~2 =
For one A/a and one a/a, we should note that this may happen in two ways. the
1/4.

/' (Success) + P (Failure) = 1* birth first or the a/a first.
P (Success) I - P (Failure) Therefore.
I - 18/20
2/20 = 1/10 = the same, result obtained directly. lJd/a. a/2) = y2 X 1/2 = I~
..£(aLq~.Jta.),~tb2~UL2 = 1~Jl.n~L
,...llWL<ca.IldJ ll.IJJJ = JL4 + 1/4 = 1/2.
*This is a usl'ful relation 10 keep in mind. Unlike the case above. it is Qnl'n easier 10 calculate the
prnhahility or I l i ohwining the desired result. and then subtra(:ting rrom unily. than to calculate
The previous line is unique because the probability of obtaining the "expected"
direrllv lhe prnhahilily or the desired result. ratio is the same as a fraction in the ratio.

., CHANCE AND THE DISTRIBUTION OF FAMILIES 183'
"1~2 TEXTBOOK OF HUMAN GENETICS

WHAT THE GENETIC RATIOS MEAN
'I
(.
This "multiplication rule" is valid whether the desired events occur simulta
neously or in a particular order. Furthermore, it is applicable whether the proba
hilities of successive events are independent or dependent. Drawing balls from the We may now consider the meaning of the Mendelian ratios.

two urns-described above-is an example of independent probabilities, where the Meaning I: :'~genelic rgti!Lslq}e~_lh(' [JrQhal!.W.!J~.t(U· (LsU!gle..bil:th. When we state

probability of one event docs not influence the probabilty of another in the series.
.A/a X a/a ....... II1JIjJl._IJ2.~q/.a

Likewise. if only 011e urn is used and the ball replaced between pulls. the probabil
ities of successive choices are independent. To illustrate dr.11£'ndent probabilittes_,e we are saying that at any given birth from these parents the odds are I: I that the
may consider only the first urn and calculate, for example, the .P!9babilit.y of pi~k child will be a/a. On the other hand. for the mating
ing.a_hIHdsJ!.~IU~ice"",iJ;uuccession, if the Jirst one rem~qye_dJs_fJQLScgO_.mJLI.1()t
I~nlacc.(lpefore pulling_o.uJ_the..se.cond. A/a X A/a ....... 1/4 A/A. 1/2 A/a, 1/4 a/a,

----------.--~-'.
fool' the first choice. as before.
the stated ratio means that at any given birth there is twice as much chance that
P (BI or B2) = 21J. the baby will be A/a than that it will be A/A. Likewise the odds are_2..:...Lfm.,..dul
_v_e[s.u~ALa and.t:J fOL!1L.:cCv.ersus~ala. IfA.is._<:!QOJi(l3tlJ..o.Y.cL(1. the probability is I
If the tirst choice is successful. only four balls remain in the urn and only one of
~/~thauuAil.dJi:Q.qUb!!Jas.tmating...w.illshow-th~...A....phcnot.y.pe. J./.;l...lb.3LiJ~
these is black. Therefore. for the second choice
sho~{L1~henotYJ!e, in other words. three times as much chance for gene :I to be
J:.Jhj~c;Jsl_=_tM_ present at least once than for its complete absence. a/a.
Meaning 2: itl,l1fJ,ILSiqAbiv.• lhe..resuiUu&qcsf.ecLby.. tb.('"gclI.cli.(.;..raUQ.ba:s..tlu· "Iot/ql
The probability that bOlh events happen is. then
probabili/..!:. This probability is usually not the probability in the Mendelian ratio.
2/5_~_1./_4 = 2/2Q....:;.....lLLO but must be calculated for each case.
The genetic ratio leads us to expect that the distribution of children in a family
The student is often disturbed by dependent probabilities because it seems
will conformJo this ratio, but we know it usually will not do so. For exam"plc. if
more necessary than for independent probabilities to sUJ)pose lhat the firSt choice
the parents in the mating
is successful beforecalcutating tlieprobability of the second. He should keep upper
most in mind that we arc seeking the likelihood that both events do happen. We A/a X a/a ....... 1/2 A/a. 1/2 a/a

wish to calculal~J!l<;..probability of SIlC(!'.!S. not the probability of failure. Failure.
not obtaining the desired result, includes all the other results that could occur: the decide to have two children the ratio "predicts" that they will have one il/a child
first choice is "right" but the second "wrong." the first wrong even if the second is and one a/a child. But our biological knowledge. and observation of actual fami
right. or both choices are wrong. Calculating the probability offailure on the same lies. tells us that such families will ~onsists of two A/a children or twoJl/a children
hasis used for thc probability of success verifies our result: as wellas one A/a and one a/a. Using our rules of probability and the first meaning
for the genetic ratio. we can calculate the probability or expected relative frequency
f> (black. white) = 2/5 X 3/4 = 6/20 of these sibships.
P (white. black) = 3/5 X 2/4 = 6/20
P (white. white) = 3/5 X 2/4 = 6/20 lX?tJLa) "" fJJ/a...::l£g) =Jj2 X 1/2 = l/.i

Since any of these alternative spells failure. Likewise,

P (Failure) = 6/20 + 6/20 + 6/20 = 18/20. J.j2a/q) = eLqLQ,.J1£q) = JJ~ X 1/2 = -1L4:
Since
For one A/a and one a/a. we should note that this may happen in two Vay~, the
/' (Success) + P (Failure) = 1* A/a birth first or the a/a first.
P (Success) = I - P (Failure) Therefore.
"" I - 18/20
= 2/20 = 1/0 = the same result obtained directly'; = 1i2._~~LL2-=--ljS..
.P'(d/.i1,-a.iJl)
p(f}ja. A/.Q.l =
l.a~>-U.L2 = 1/..4. and
./
L'<L:!/a and I a/!!..~ =:..1/4 + 1/4 = t.Lb
*This is a useful rcialion to keep in mind. Unlike the case above, it is often ('asier 10 calculale the The previous line is unique because the probability of obtaining the "cxpceted"
prohahility of I/O/ obtaining the dcsir('d rcsult. and Ihen subtracting from unity. than to calculatc
dirl:l'lly tht' prnhahility of the desired result. ratio is the same as a fraction in the ratio.

l~O '~

seen already that l::-e ignore the probability of new mutations when we predict that Urn A Urn B Urn A Urn B UrnA Urn Il Urn A Urn B UrnA Urn B

thc cross: ii8 0 fl0 0 8G) 00 GQ
~~"""'aIlA/A. "1100~00 00 00 80
In rrobability terms we are saying
-ott 0 @ -.1 0 8@ G)0 80
P lJ/A X A/A ....... A{"-L::'--l.

80 0 80 G)0 80
and Ihis cross is therefore usually written

.1/..1 X A/A ....... J A/A.

88 08 G8 G8 G8
Implicit in asking the probability of obtaining a while ball from the urn is the 08 08 8 0) 08
idea that it docs not matter whether we obtain WI. W2. or W3. Any of Ihese alter Fig. 8-1 The 30 combinations possible when a ball is drawn from each of two urns
natin- CTnts spells success. Thus. the first urn contains 2 black balls and 3 white balls, the second contains 3 black onl
and 3 white ones, and the balls are equal in size but distinguishable by number.
I' (white) :: I + I + I
5

1/5 + 1/5 +
Thus,
Aul.

2X3
P(W!) = 1/5.
P (black, black) = 5 X 6
P (W2) = 1/5. and
2 3
P(W3) = 1/5.
= 5X 6'
IicncC'.
But.
I'(while) P(WI) + P(W'l) + I'(W3).
2
P (black from urn A)
In other 'ords:..i.fall erelll has sereral allf'!1l.S!.!.i.!,t.lill!2IS and allainQl£I1J •o C'lny 5
ill!l:UEltixl,~is~(~(msidcreJ:LaJtiljnmcr!t..Qf.lhc..desired>e_vcnl. the J!f'Jh(//JiliO' IJj'SIIC<'l''
and
IS (II(' SUIIl o[lhe pro/'ahililies of'lliese allemalire Ji}f!Ul. In commonsense lerms.
hell we arc not particular or choosy the chances for Success arc increased.
.-n equally interesling probability is that of a success which is composed of
~ P (black from urn 8).
sewal en~nts that must happen simultaneously or in succession. Clearly. Ihis
probahilily must be smaller than the likelihood of attaining anyone of the events Thus. the probability of obtaining black balls from both urns is the product 0
alone. How much smaller'! To sol'(' this problem let us relurn to the urn wilh Ihe Ihe probabilities of obtaining a black ball from cach urn.
two hlack halls and the three white ones and set up a second urn with Ihree black We may generalize this into a rule:.:rhf.•J!rohahilill' (!( ol>ta;I/;l/g cl'rtoiJl.:Ji!PU
halls. A3. R4. R5. and three white ones. W4. W5. and W6. If a ball is to be removed tal/eOIlS (or suq::£l>siyc,) _('n'llls is II/(' prodl~<.!_!f Ille pmhahilifil's 0" Ihe l'r(!}fi
From each urn. how likely is it. for example. that both will be black? To answer Ihis Iiimln,:d.To check this generalization. ;"'c may note that
c may return to our basic concept of probability and enumerate how many
. .) 3 3
l'qually probable events. each consisting of one ball from urn A and one ball from P (whIte. whIte = 5 X "6
urn H. can happen and note how many of these consiSI of a black ball from each
9
urn. Thl' equally probable events arc shown in Fig. 8-1.
30
There arc 30 equally probable events and 6 of these represent pulls of two black
I Ills. Thus.
and Ihis corresponds to the enumerated proportion.
I'(black. black) = 6/30 This rule too. fits our commonsense expectations. The likelihood Ihat both wil
happen must be smaller than the likelihood ofeilher one alone. for each c .wld 011L'1
Note Ihat the )0 in the denominator is the product of the number of balls in urn happen without the other. Since probabilities arc fractions less than one. nlultipl)
i , and the number of balls in urn R: likewise. the 6 in the numeralor is Ihe product ing them has the effect of producing a still smaller fraction. that is. dccreasing 1h
(11' thl' numher of black balls in urn A and the number of hlacks in urn R.
probabilitv. .

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