Download Math 141 Lecture Notes

Math 141 Lecture Notes
Section 7.5 Conditional Probability and Independent Events
Conditional Probability
The probability of an event occurring given that another even has already occurred is a
conditional probability.
Number of elements in A  B n( A  B)
P( B | A) 

Number of elements in A
n( A)
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( A  B)
P( B | A) 
P( A)
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?
Example 2: A pair of fair dice is cast. What is the probability that the sum of the
numbers falling uppermost is 7 if it is known that one of the numbers is a 5?
Example 3: In a test conducted by the U.S. Army, it was found that of 1000 new
recruits, 600 men and 400 women, 50 of the men and 4 of the women were red-green
color-blind. Given that a recruit selected at random from this group is red-green colorblind, what is the probability that the recruit is a male?
Product Rule:
P( A  B)  P( A)  P( B | A)
P( E  F  G )  P( E )  P( F | E )  P(G | E  F )
Example 4: There are 300 seniors in Jefferson High School, of which 140 are males. It
is known that 80% of the males and 60% of the females have their driver’s license. If a
student is selected at random from this senior class, what is the probability that the
student is
a. A male and has a driver’s license?
b. A female who does not have a driver’s license?
Example 5: Two cards are drawn without replacement from a well-shuffled deck of 52
playing cards. What is the probability that the first card drawn is an ace and the second
card drawn is a face card?
More on Tree Diagrams
A finite stochastic process is an experiment consisting of a finite number of stages in
which the outcomes and associated probabilities of each stage depend on the outcomes
and associated probabilities of the preceding stages.
Example 6: Two cards are drawn without replacement from a well-shuffled deck of 52
playing cards. What is the probability that the second card drawn is a face card?
Example 7: The picture tubes for the Pulsar 19-inch color television sets are
manufactured in three locations and then shipped to the main plant of Vista Vision for
final assembly. Plants A, B, and C supply 50%, 30%, and 20%, respectively, of the
picture tubes used by the company. The quality-control department of the company has
determined that 1% of the picture tubes produced by plant A are defective, whereas 2%
of the picture tubes produced by plants B and C are defective. What is the probability
that a randomly selected Pulsar 19-inch color television set will have a defective picture
tube?
Example 8: A box contains eight 9-volt transistor batteries, of which two are known to be
defective. The batteries are selected one at a time without replacement and tested until a
nondefective one is found. What is the probability that the number of batteries tested is
(a) one, (b) two, and (c) three?
Independent Events
In general, two events A and B are independent if the outcome of one does not affect the
outcome of the other.
Independent Events:
If A and B are independent events, then P( A | B)  P( A)
and
P( B | A)  P( B) .
Test for the Independence of Two Events
Two events A and B are independent if and only if P( A  B)  P( A)  P( B) .
Note: Do not confuse independent events with mutually exclusive events. The former
pertains to how the occurrence of one event affects the occurrence of another event,
whereas the latter pertains to the question of whether the events can occur at the same
time.
Example 9: Consider the experiment consisting of tossing a fair coin twice and observing
the outcomes. Show that the event of “heads” in the first toss and “tails” in the second
toss are independent events.
Example 10: A survey conducted by an independent agency for the National Lung
Society found that of 2000 women, 680 were heavy smokers and 50 had emphysema.
Of those who had emphysema, 42 were also heavy smokers. Using the data in this
survey, determine whether the events “being a heavy smoker” and “having emphysema”
are independent events.
Independence of More Than Two Events
If E1 , E 2 ,..., E n are independent events, then
P( E1  E2  ...  En )  P( E1 )  P( E2 )  ...  P( En )
Note: The converse of the theorem is not necessarily true.
Example 11: It is known that the three events A, B, and C are independent and
P(A) = .2, P(B) = .4, and P(C) = .5. Compute:
a. P(AB)
b. P(ABC)
Example 12: The Acrosonic model F loudspeaker system has four loudspeaker
components: a woofer, a midrange, a tweeter, and an electrical crossover. The qualitycontrol manager of Acrosonic has determined that on the average 1% of the woofers,
0.8% of the midranges, and .5% of the tweeters are defective, while 1.5% of the electrical
crossovers are defective. Determine the probability that a loudspeaker system selected
at random coming off the assembly line and before final inspection is not defective.
Assume that the defects in the manufacturing of the components are unrelated.