Probability is the branch of mathematics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur.[note 1][1][2] The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ('heads' and 'tails') are both equally probable; the probability of 'heads' equals the probability of 'tails'; and since no other outcomes are possible, the probability of either 'heads' or 'tails' is 1/2 (which could also be written as 0.5 or 50%). These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in areas of study such as statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.

1. M.Sc MICROBIOLOGY
SEMESTER – II
BIOSTATISTICS
TOPIC-PROBABILITY

2. PROBABILITY
CONTENT:-
Introduction
Important terms and concepts
Basic features of probability
Types of probability
Theorems on probability
Conditional probability

3. DEFINITION:-
Probability represents the chance of occurrence with which an event is expected to
occur on an average.
If an event can occur in N mutually exclusive and equally likely ways, and if n of
these possess a characteristics,the probability of occurrence of E is equal to
p(E)= n/N
where,
N= Represents the number of trials
X or n= Represents the number for which the event has occurred
p= Represents the probability
In all research program medicines trial and treatments,scientist take the help of
probability to reach any conclusion.

4. Relative frequency of probability :-
The relative frequency of probability depends on the repeatability
of the event(E) and the ability to count the number of repetitions.
If some trials is repeated a large number of times(N) and if some
resulting event(E) occurs m times, the relative frequency of E is
m/N.
P(E) = Frequency of occurrence of an event
Total number of trials
= m/N

5. Example:
Microbiologist wants to estimate the probability of getting gram
positive bacteria by preparing gram staining slide.
She prepared 10 slides and observed gram positive bacteria in 6
slides.
Relative Frequency=
6
10
=
3
5
=0.6

6. Probability In Biology:-
The event of probability is basic to
many priciples of classical genetics
example segregation, independent
assortment and epistasis etc.
In Mendelian genetics, the appearance
of different traits in offspring is by
chance. Though the approximate
probability ratios can be worked out
by genecists, the exact number of
offsprings for any given feature cannot
be predicted accurately.

7. Important Terms And Concepts
• Experiment:- A process which results in some well defined outcome is known as
an experiment.
For example- When a coin is tossed, we shall be getting either a head or a tail i.e.
its outcome is a head or a tail, which is well defined.
• Random experiment or Trial:- Random experiment means all the outcomes of the
experiment are known in advance, but any specific outcome of the experiment is not
known in advance. It is also called as trial.
Example – Tossing of a coin is a random experiment because there are only two
possible outcomes,head and tail, and these outcomes are known well in advance. But
the specific outcome of the experiment i.e. whether a head or tail is not known in
advance.

8. • Sample space and Sample point:- A set of all possible outcomes from an experiment is
called a sample space. It symbol is S and the number of components is denoted by n (s).
The component of sample space are known as sample points.
Example- Toss a coin the sample space is S = {H,T}
• Events:- The possible outcomes of experiments are called events.
Example- On tossing a coin,the possible outcomes is a head(H) or tail(T).Here getting a head
or a tail is an event of the experiment.
Events can be of following different types:-
1. Simple events- An event consisting of only one sample point is called a simple event.
For example- (i) When a dice is rolled once and the event that shows 5,it is a simple event.
(ii) Birth of a male or a female child in a family is a simple event.

9. 2. Mixed or Joint events- If two or more simple events occur simultaneously ,they represent
mixed events .These represent happening of two events together.
Example- there is a bag containing 5 white balls and 5 red balls.Now on making a draw of 2
balls at random, the events that both are white or both are red or one is white and one is red are
compound events.
3. Independent event- When the occurrence of one event doesnot affect the occurrence of
another event, it is said to be an independent event.
Example- When a coin is tossed twice, the result of second toss would in no way be affected
by the result of the first toss.
4. Dependent event- When the occurrence of one event influences the occurrence of the other
event the second event is said to be dependent event.
Example- When a person draws a card from a full pack and doesnot replace it, the result of the
draw made afterwards will be dependent on the first draw.
5. Exhaustive event- It is the total number of all the possible outcomes of an experiment.
Example- When a dice is thrown, there are 6 possible outcomes i.e. 1,2,3,4,5,6. All these
outcomes are termed as exhaustive events.

10. 6. Mutually exclusive events- When occurrence of one event preludes the possibility of
occurrence of another event, two events are disjoint events and are called mutually exclusive
events.
It means both events cannot occur simultaneously and the occurring of one doesnot allow the
occurrence of the other event.
Example- In tossing a coin, the events head and tail are mututally exclusive. There will be
either head or tail.
7. Equally likely events- If one of the event cannot be taken in preference to the other, the
two events are considered equally likely events.
Example- In tossing of a coin the chance of getting a head is the same as that of getting a tail.
Therefore the two events of getting a head or a tail are equally likely events.
8. Null or Impossible events- An event having no sample point i.e. no possibility of
occurrence is called a null event. It is denoted by phi (ᶲ).
Example- Probability of getting 7 in a throw of dice is a null event because dice doesnot have
sample point 7.

11. Basic Features Of Probability
Following basic laws of probability are most specific:-
1. If probability of occurrence of an event is 1,the event will occur
certainly.
2. If probability of occurrence of an event is 0, the event will never
occur.
3. The probability of any event must assume a value between 0 and 1.
4. The sum of probabilities of all the simple event in a sample space
must be equal to 1. it can also be said that the probability of the
sample space in any experiment is always one.

12. Types Of Probability
There are two types of probability:-
1. Classical or Mathematical:- When the likelihood of occurrence of various chance
event is based on our previous knowledge it is called theoretical or classical
probability. It is the ratio of number of favourable cases.
Explanation- If an experiment has n mutually exclusive, equally likely and exhaustive
cases out of which m are favourable to the happening of the event (E), the probability
of the happening of E is denoted by P(E) and is represented by
P(E)=m/n=
𝑛𝑜.𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
𝑇𝑜𝑡𝑎𝑙 𝑛𝑜.𝑜𝑓 𝑐𝑎𝑠𝑒(𝑚𝑢𝑡𝑢𝑎𝑙𝑙𝑦 𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑣𝑒,𝑒𝑞𝑢𝑎𝑙𝑙𝑦 𝑙𝑖𝑘𝑒𝑙𝑦 𝑎𝑛𝑑 𝑒𝑥ℎ𝑎𝑢𝑠𝑡𝑖𝑣𝑒 𝑐𝑎𝑠𝑒𝑠
The probability of the event that doesnot happen is represented as,
P(Ē)=
𝑁𝑜.𝑜𝑓 𝑐𝑎𝑠𝑒𝑠 𝑢𝑛𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑡𝑜 𝑒𝑣𝑒𝑛𝑡 (𝑛−𝑚)
𝑇𝑜𝑡𝑎𝑙 𝑛𝑜.𝑜𝑓 𝑐𝑎𝑠𝑒𝑠(𝑛)
P(E)+P(Ē)=
𝑚
𝑛
+
(𝑛−𝑚)
𝑛
= 1

13. Example-
Q. Find the probability of getting an even number in a single throw with a dice?
Solution- The possible outcome cases in the throw of a dice are six 1,2,3,4,5,6.
Here, favourable cases are 2,4,6 and these are three in number.
Probability of getting an even number = 3/6 =1/2

14. Salient Features Of Classical Probability
From the above definition it is clear that,
a. The probability of an event is the ratio of the number of favourable cases to the
exhaustive cases in a trial.
b. The probability of an event which can occur is 1.
c. The probability of an event which cannot occur is 0.
d. The sum of the probabilities of happening and not happening of an event is always
equal to 1.

15. Limitation of Classical Probability:-
a. Classical probability is based on the feasibility of grouping the possible outcomes
of the experiments into mutually exclusive, exhaustive and equally likely events.
b. Classical probability is applicable only when sample space is finitely enumerable.
c. Classical probability fails to give required probability when number of possible
outcome is infinitely large.
d. Classical probability is not applicable to cases where outcomes of an experiment
cannot be enumerated.

16. 2. Statistical or Empirical Probability:-
Statistical probability is the relative frequency of occurrence of an event in a large number of
trials.
P(E)= limit 𝑛 ∞ m/n
For example- Almost all probabilities in the health field and ascertaining the effect of a
medicine are derived empirically.Like the probability of survival for a cancer patient is
assumed based on the known survival rates of similar patients who have same stage of the
cancer and have undergone the same treatment.
Salient Features :-
i. The probability is determined by the measurement (experiment).
ii. Measure frequency of occurrence.
iii. Not all outcomes necessarily have equal probability.

17. Theorems On Probability
There are two important theorems of probability, namely:-
1. Addition theorem for total probability.
2. Multiplication Theorem or Theorem on Compound
Probability.

18. 1. ADDITION THEOREM OR SUM RULE OF PROBABILITY :-
* When events are mutually exclusive:-
The probability of occurrence of one of two mutually exclusive events is the sum of total of
their individual probability.
P(A ∪ B) = P(A) + P(B)
Example- if the probability of the horse A winning the race is 1/5 and theprobability of horse
B winning the same race is 1/6. What would be the probability of that one of the horses will
win the race?
Solution- Probability of winning of horse A= 1/5
Probability of winning of horse B= 1/6
P(A ∪ B) = P(A) + P(B)
=1/5+1/6= 11/30 or 0.367
The probability of A or B winning the race is 0.367 or 11/30

19. * When events are not mutually exclusive:-
The probability of occurrence of event A or event B is equal to the probability that event A
occurs plus the probability that event B occurs, minus the probability that both A and B occur
simultaneously.
P(A ∪ B) = P(A) + P(B)-P(A∩ B)
Example- Probability of king and heart in a pack of card?
Solution- Total cases are 52
Number of king= 4
Probability of king= 4/52
Number of heart= 13
Probability of heart=13/52
Probability of king of heart= 1/52
P(A ∪ B) = P(A) + P(B)-P(A∩ B)
=
4
52
+
13
52
−
1
52
=
16
52

20. 2. Multiplication Theorem:-
Conditional Probability :-A conditional probability is the probability of an event
occurring, given that another event has already occurred.
The conditional probability of event B occurring, given that event A has occurred, is
denoted by P(B/A) and is read as “probability of B, given A.”
Similarily, P(A/B) is conditional probability of A given that B has already happened.
Formula:- P(A/B)=
(𝑨∩𝑩)
𝑷(𝑨)
P(B/A)=
(𝐀∩𝐁)
𝐏(𝐁)

21. *When events are independent:-
The probability of two or more independent events occuring together is the
product of the probabilities of the occurrence of individual events.
P(A∩ B)= P(A) . P(B)
Example:- If five coins are tossed,what is the probability that all will show a
head?
Solution:- Let P(A) denotes the probability of showing a head,
P(A)=1/2
Now probability of showing a head when 5 coins are tossed=
1
2
×
1
2
×
1
2
×
1
2
×
1
2
=
1
32

22. *When events are dependent:-
If the probability of occurring of event B is dependent on the occurrence of event A,the
probability of occurrence of events A and B can be given as
P(A∩ B)= P(A)× 𝑃(
𝐵
𝐴
)
OR
P(A∩ B)= P(B)× 𝑃(
𝐴
𝐵
)
EXAMPLE:-If we draw a card from a full pack of card then again draw another card from the
same pack of card then,what is the probability of gettingAce and King?
Total no. of case=52
P(Ace)= 4/52
P(King)=4/51
P(A∩ B)=4/52 × 4/51
=ans

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