Math 141: Business Mathematics I
Fall 2015
§7.5 Conditional Probability and Independent Events
Instructor: Yeong-Chyuan Chung
Conditional Probability
In general, the probability of an event is affected by the occurrence of other events and/or
by the knowledge of information relevant to the event because the sample space changes.
Example. 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?
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§7.5 Conditional Probability and Independent Events
The probability of an event occurring given that another event has already occurred is what
we call a conditional probability. If A and B are two events, we write P (B|A) for the
conditional probability of B given A.
Note: P (B|A) is different from P (A|B).
In the previous example, we saw that the sample space was reduced to a smaller one after
the first card had been drawn. More generally, when we are considering the conditional
probability P (B|A), the condition that event A has occurred means that the sample space
has been reduced to A. With respect to this reduced sample space, the outcomes in event
B are precisely those in A ∩ B. This gives us the following formula for computing P (B|A):
If P (A) 6= 0, then P (B|A) =
P (A ∩ B)
.
P (A)
We may rewrite the formula to give us the following product rule:
P (A ∩ B) = P (A) · P (B|A).
This product rule can be useful if we know the conditional probability of B given A, and we
want to compute the probability of A and B occurring.
The product rule can be understood by means of a tree diagram:
Example. At a party, 31 of the guests are women. 75% of the women wore sandals and 40%
of the men wore sandals. What is the probability that a person chosen at random at the party
is a man wearing sandals?
§7.5 Conditional Probability and Independent Events
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Example (Exercise 16 in the text). A pair of fair dice is rolled. Let E denote the event that
the number falling uppermost on the first die is 4, and let F denote the event that the sum
of the numbers falling uppermost is 6.
(a) Compute P (F ).
(b) Compute P (E ∩ F ).
(c) Compute P (F |E).
(d) Compute P (E).
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§7.5 Conditional Probability and Independent Events
Example (Exercise 40 in the text). Figures obtained from a city’s police department seem
to indicate that of all motor vehicles reported as stolen, 64% were stolen by professionals,
whereas 36% were stolen by amateurs. Of the vehicles presumed stolen by professionals, 24%
were recovered within 48 hr, 16% were recovered after 48 hr, and 60% were never recovered.
Of the vehicles presumed stolen by amateurs, 38% were recovered within 48 hr, 58% were
recovered after 48 hr, and 4% were never recovered.
(a) Draw a tree diagram representing these data.
(b) What is the probability that a vehicle reported as stolen is stolen by a professional and
will be recovered within 48 hr?
(c) What is the probability that a vehicle stolen in this city will never be recovered?
§7.5 Conditional Probability and Independent Events
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Independent Events
We have been considering situations in which the occurrence of one event affects the occurrence of another event. There are also situations in which the occurrence of one event has no
effect on the occurrence of another event. For example, consider the experiment of tossing
a coin twice. If H is the event that the first toss produces heads, and T is the event that
the second toss produces tails, then intuitively the events H and T are independent of each
other because the outcome of the first toss does not affect the outcome of the second toss.
We say that two events A and B are independent if P (A|B) = P (A) and P (B|A) = P (B).
The first equality says that knowing that B has occurred does not change the probability of
A occurring. The second equality says that knowing that A has occurred does not change
the probability of B occurring.
By applying the product rule that we saw earlier, we see that two events A and B are
independent if and only if
P (A ∩ B) = P (A) · P (B).
This is what we often use to test whether two events are independent or not.
Example (Exercise 6 in the text). If P (A) = .6, P (B) = .8, and P (A ∩ B) = .2, are A and
B independent events?
Example (Exercise 8 in the text). If P (Ac ) = .3, P (B c ) = .4, and P (A ∩ B) = .42, are A
and B independent events?
Example (Exercise 10 in the text). If A and B are independent events, P (A) = .35, and
P (B) = .45, find
(a) P (A ∩ B)
(b) P (A ∪ B)
(c) P (A|B)
(d) P (Ac ∪ B c )
§7.5 Conditional Probability and Independent Events
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Miscellaneous examples
Example (Exercise 24 in the text). Five black balls and four white balls are placed in an
urn. Two balls are then drawn in succession. What is the probability that the second ball
drawn is a white ball if:
(a) The second ball is drawn without replacing the first?
(b) The first ball is replaced before the second is drawn?
Example (Exercise 42 in the text). The probabilities that the three patients who are sched1
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uled to receive kidney transplants at General Hospital will suffer rejection are 12 , 13 , and 10
Assuming that the events (kidney rejection) are independent, find the probability that:
(a) At least one patient will suffer rejection.
(b) Exactly two patients will suffer rejection.
§7.5 Conditional Probability and Independent Events
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Example (Exercise 46 in the text). The admissions office of a private university released
the following data for the preceding academic year: From a pool of 3900 male applicants,
40% were accepted by the university, and 40% of these subsequently enrolled. Additionally,
from a pool of 3600 female applicants, 45% were accepted by the university, and 40% of these
subsequently enrolled. Based on these data, what is the probability that:
(a) A male applicant will be accepted by and subsequently will enroll in the university?
(b) A student who applies for admissions will be accepted by the university?
(c) A student who applies for admission will be accepted by the university and subsequently
will enroll?
Example (Exercise 52 in the text). Two cards are drawn without replacement from a wellshuffled deck of 52 cards. Let A be the event that the first card drawn is a heart, and let B
be the event that the second card drawn is a red card. Show that the events A and B are
dependent events.