Chapter 6.3C
1. In 2000, the International Revenue Service received 129,075,000 individual tax returns. Of
these, 10,855,000 reported an adjusted gross income of at least $100,000, and 240,000
reported at least $1 million. If you know that a randomly chosen return shows an income of
$100,000 or more, what is the conditional probability that the income is at least $1 million?
P(at least $100,000|at least $100,000) = 0.0019/0.0841 = 0.0226
******There is a type on your homework here. It should say over $100,000, not over 1 million.
2. A shipment contains 10,000 switches. Of these, 1000 are bad. An inspector draws switches at
random, so that each switch has the same chance to be drawn.
(a) Draw one switch. What is the probability that the switch you draw is bad? What is the
probability that it is not bad? P(Bad) = 1000/10000 = .1, P(not bad) = 0.9.
(b) Suppose that the first switch drawn is bad. How many switches remain? How many of them
are bad? Draw a second switch at random. What is the conditional probability that this
switch is bad?
9,999 switches remain, 999 are bad. P(second is bad|first is bad) = 999/9,999 = 0.099991
(comment: Knowing the result of the first trial changes the conditional probability for the
second trial, so the trials are not independent. But because the shipment is large, the
probabilities change very little. The trials are almost independent.)
3. Here are the counts (in thousands) of earned degrees in the United States in the 2005-2006
academic year, classified by level and by the sex of the degree recipient:
Bachelor’s
Master’s
Professional
Doctorate
Total
Female
784
276
39
20
1119
Male
559
197
44
25
825
Total
1343
473
83
45
1944
(a) If you choose a degree recipient at random, what is the probability that the person you
choose is a woman?P(Female) = 1119/1944 = .576
(b) What is the conditional probability that you choose a woman, given that the person chosen
received a professional degree? P(Female | professional degree) = 39/83 = 0.4699
(c) Are the events “choose a woman” and “choose a professional degree recipient”
independent? How do you know?
No. If the events A = choose a woman and B = Professional degree are independent, then
P(Female) = P(Female|Professional). We know from the previous problems that P(female)=.576
and P(Professional degree) = .4699, so not independent.
(d) What is the probability that a randomly chosen degree is a man?
P(Male) = 1343/1944 = 0.4244
(e) What is the probability that the person being chosen received a bachelor’s degree, given
that he is a man? P(bachelor’s | Male) = 559/825 = .6776
(f) Use the multiplication rule to find the joint probability of choosing a male bachelor’s degree
recipient. Check your result by finding this probability directly from the table of counts.
P(Male)*P(Bachelors|Male) = (0.4244)(.6776) = 0.2876. If we look at the table, there are 559/1944 =
.2876, so yes, they do agree.