1. Independent and Dependent Events
The student is able to (I can):
• Identify whether two (or more) events are independent or
dependent
• Find the probabilities of independent or dependent events
2. independent eventsindependent eventsindependent eventsindependent events – two events in which the occurrence of
one event does not affect the occurrence of the other
event. Symbolically, this is written:
dependent eventsdependent eventsdependent eventsdependent events – two events in which the occurrence of
one event does affect the occurrence of the other
event.
Terms that indicate dependence:
• “without replacement”
• “without returning it”
( ) ( ) ( )= iandP A B P A P B
3. Example: Determine if events A and B are independent.
1.
2.
( ) ( ) ( )
3 3 3
, , and
5 10 25
P A P B P A B= = =
( ) ( ) ( )
1 2 1
, , and
4 5 10
P A P B P A B= = =
4. Example: Determine if events A and B are independent.
1.
Since this does not equal P(A and B),
the events are dependent.
2.
Since this doesdoesdoesdoes equal P(A and B), the
events are independent.
( ) ( ) ( )
3 3 3
, , and
5 10 25
P A P B P A B= = =
3 3 9
5 10 50
=i
( ) ( ) ( )
1 2 1
, , and
4 5 10
P A P B P A B= = =
1 2 2 1
4 5 20 10
= =i
5. 3. As part of an experiment, you are rolling a six-sided die
and spinning a spinner that is divided into 5 evenly-
spaced sections
What is the probability that
you would roll a four and
spin red?
6. 3. As part of an experiment, you are rolling a six-sided die
and spinning a spinner that is divided into 5 evenly-
spaced sections
What is the probability that
you would roll an even
number and spin red?
( ) ( ) ( )and
3 1
6 5
1
10
P A B P A P B=
=
=
i
i
7. conditional probabilityconditional probabilityconditional probabilityconditional probability – the probability that event B occurs
given that event A has occurred. This is written as
and read “B given A”. To find the probability of
dependent events A and B :
Example: Find the missing probability
( )|P B A
( ) ( ) ( )and |P A B P A P B A= i
( ) ( ) ( )
63 9
and , | , ?
200 20
P A B P B A P A= = =
8. conditional probabilityconditional probabilityconditional probabilityconditional probability – the probability that event B occurs
given that event A has occurred. This is written as
and read “B given A”. To find the probability of
dependent events A and B :
Example: Find the missing probability
( )|P B A
( ) ( ) ( )and |P A B P A P B A= i
( ) ( ) ( )
63 9
and , | , ?
200 20
P A B P B A P A= = =
( )
63
1260 7200
9 1800 10
20
P A = = =
9. Examples:
1. A bag contains twenty $1 bills and five $100 bills. You
randomly draw a bill from the bag, set it aside, and then
randomly draw another bill from the bag. What is the
probability that you drew two $100 bills?
( )
( )
5
25
4
|
24
P A
P B A
=
= If you drew one $100 bill, there are now
four left. There’s also one fewer bill.
( )
5 4 1
and B
25 24 30
P A = =i
10. 2. You are drawing cards from a randomly-shuffled,
standard deck of cards. What is the probability that you
would draw three hearts when (a) you replace each card
before selecting the next card, and (b) you do not replace
each card before selecting the next card?
a) Because you replace the card each time, the events
are independent.
b) Because you are notnotnotnot replacing the cards, the events
are dependent.
13 13 13 1
52 52 52 64
P = =i i
13 12 11 11
52 51 50 850
P = =i i
