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Math 1313
Section 6.5
Conditional Probability and Independent Events
Example 1: Two cards are drawn without replacement from a well-shuffled
deck of 52 playing cards.
a. What is the probability that the first card drawn is an ace?
b. What is the probability that the second card drawn is an ace given that the
first card drawn was not an ace?
c. What is the probability that the second card drawn is an ace given that the
first card drawn was an ace?
Note that b., c. are both conditional probabilities.
Conditional Probability of an Event
If A and B are events in an experiment and P(A) ≠ 0, then the conditional
probability that the event B will occur given that the event A has already
occurred is
P( B | A ) =
P( A I B )
P( A )
Example 2: Given P(E) = .26 , P(F) = .58 , and P(E I F) = .02
Find P(E|F)=
P(F|E)=
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Math 1313
Section 6.5
=.
Example 3: Given
| .
Find
,
∩
=.
∪
and
=.
.
Given the Table:
A
AIB
B
c
B IA
Total
B I Ac
c
B IA
B
c
A
U
Example 4: Records about rainfall and the weather forecasts for a
region in west Texas have been summarized in the following
table:
Forecast of Rain
Forecast of No Rain
Rain
66
14
Sum
80
No Rain
156
764
920
Sum
222
778
1000
a. Calculate the probability that it rained, given that the forecast called for
rain.
b. Calculate the probability that it did no rain, given that the forecast
called for no rain.
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Math 1313
Section 6.5
Example 5: A pair of fair dice is cast. What is the probability that the sum of
the numbers falling uppermost is 6, if it is known that exactly one of the
numbers is a 2?
Example 6: A pair of fair dice is cast. What is the probability that at least one
of the numbers falling uppermost is a 5, if it is known that the two numbers
are different?
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Math 1313
Section 6.5
Example 7: An urn contains 4 blue and 7 green marbles. Two marbles are
drawn in succession without replacement from the urn.
a. What is the probability that the first marble is blue?
b. What is the probability that the second one is blue, given that the first one
was blue?
c. What is the probability that the second marble is green?
d. What is the probability that they are both the same color?
Product Rule
Suppose we know the conditional probability and we are interested in finding
P ( A I B)
Then if P ( A ) ≠ 0 then P ( A I B) = P ( A ) P ( B | A ) .
P( B | A ) =
Where did this come from:
P(A I B)
P(A) and we solved for P ( A I B) .
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Math 1313
Section 6.5
Example 8: could be done with replacement (Notice the difference with on the
tree diagram) this will help with Quiz 9 # 9 and #10:
An urn contains 4 blue and 7 green marbles. Two marbles are drawn in
succession with replacement from the urn.
Example 9: Urn 1 contains 3 white and 3 blue marbles. Urn 2 contains 5
white and 6 blue marbles. One of the two urns is chosen at random with a .58
probability of choosing Urn 2 over Urn 1. An urn is selected at random and
then a marble is drawn from the chosen urn.
a. What is the probability that the drawn marble is blue if it is known that
Urn 1 was chosen?
b. What is the probability that Urn 2 is chosen and that a white marble is
drawn?
c. What is the probability that the marble drawn from the chosen urn is blue?
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Math 1313
Section 6.5
Example 10: Beauty Girl, a cosmetics company, estimates that 34% of the
country has seen it’s commercial and if a person sees it’s commercial, there is
a 5% chance that the person will not buy its product. The company also
claims that if a person does not see its commercial, there is a 21% chance that
the person will buy its product.
a. What is the probability that a person chosen at random in the country
will not buy the product?
b. What is the probability that a person chosen at random in the country
has not seen the commercial and did buy the product?
c. What is the probability that a person chosen at random in the county
did not buy the product given the person has seen the commercial?
Example 11: A recent survey of 1000 military recruits, 600 men and 400
women, showed that 62 men were color bind and 7 women were color blind.
a. The selected recruits are chosen at random from this group find the
probability the recruit is color blind, given that the recruit is female?
b. Find the probability that a recruit chosen random is female and color
blind?
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Math 1313
Section 6.5
Independent Events
Two events A and B are independent if the outcome of one does not affect the
outcome of the other.
Test for the Independence of Two Events
Two events, A and B, are independent if and only if P( A I B ) = P( A ) • P(B ) .
(This formula can be extended to a finite number of events.)
Independent and mutually exclusive does not mean the same thing.
For example, if P( A ) ≠ 0 and P(B ) ≠ 0 and A and B are mutually exclusive
then
P( A ) • P( B ) ≠ 0
P( A I B ) = 0
so P( A ) • P( B ) ≠ P( A I B )
Example 12: Determine if the stated events are independent. The experiment
is drawing a card from a well shuffled deck of 52 playing cards.
a. A = the event of drawing as face card
B = the event of drawing a heart
b. C = the event of drawing a heart
D = the event of drawing a club.
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Math 1313
Section 6.5
The test for independence can be extended to any numbers of events. So
, , , … ,
are independent if and only if
∩ ∩
∩ …∩
∗
∗
∗ …∗
.
=
Example 13: The Ace Copy Store has four photocopy machines A, B, C, and
D. The probability that a given machine will break down on a given day is:
P(A) =
, P(B) =
, P(C) =
and P(D) =
Assuming independence what is the probability that none of the machines will
break down?
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