11.5 Independent and Dependent Events

Identify whether events are independent or depdendent
Find the probability of independent and dependent events

1. Independent and Dependent Events
The student is able to (I can):
• Identify whether two (or more) events are independ ent or
dependent
• Find the probabilities of independent or dependent events

2. independent events independent events independent events independent events–two events in which the occurren ce of
one event does not affect the occurrence of the oth er
event. Symbolically, this is written:
dependent events dependent events dependent events dependent events–two events in which the occurrence of
one event doesaffect the occurrence of the other
event.
Terms that indicate dependence:
• “without replacement”
• “without returning it”
(
)
(
)
(
)
=
i
and
P A B P A P B

3. Example: Determine if events Aand Bare 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 Aand Bare independent.
1.
Since this does not equal P(Aand B),
the events are dependent.
2.
Since this does doesdoes doesequal P(Aand 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-s ided die
and spinning a spinner that is divided into 5 evenl y-
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-s ided die
and spinning a spinner that is divided into 5 evenl y-
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 probability conditional probability conditional probability conditional probability–the probability that event Boccurs
given that event Ahas occurred. This is written as
and read “Bgiven A”. To find the probability of
dependent events Aand 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 probability conditional probability conditional probability conditional probability–the probability that event Boccurs
given that event Ahas occurred. This is written as
and read “Bgiven A”. To find the probability of
dependent events Aand 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 7
200
9
1800 10
20
P A
= = =

9. Examples:
1. A bag contains twenty $1 bills and five $100 bill s. You
randomly draw a bill from the bag, set it aside, an d then
randomly draw another bill from the bag. What is t he
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 th at you
would draw three hearts when (a) you replace each c ard
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 event s
are independent.
b) Because you are not notnotnotreplacing 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