6.22 DO HUSBANDS DO THEIR SHARE? The New York Times August 21, 1989) reported a
poll that interviewed a random sample of 1025
women. The married women in the
sample were asked whether their husbands
did their fair share of household chores.
Here are the results
Outcome
Probablilty
Does more than his fair share
0.12
Does his fair share
0.61
Does less than his fair share
These proportions are probabilities for the random phenomenon of choosing a mar-
ried woman at random and asking her opinion.
(a) What must be the probability that the woman chosen says that her husband does
less than his fair share? Why?,
(b) The event "I think my husband does at least his fair share" contains the first two
outcomes. What is its probability?
6.23 ACADEMIC RANK Select a first-year college student at random and ask what his or
her academic rank was in
high school. Here are the probabilities, based on proportions
from a large sample survey of first-
year students:
Rank:
Probability
Top 20% Second 20% Third 20% Fourth 20% Lowest 20%
0.41
0.23
0.29
0.06
0.01
(a) What is the sum of these probabilites? Why do you expect the sum to have this value?
(
b) What is the probability that a randomly
chosen first-year college student was not
in the top 20% of his or her high school class?
(c) What is the probability that a first-year student was in the top 40% in high school?

(a) To find the probability that a woman chosen says her husband does less than his fair share, we know that the sum of all probabilities must equal 1. Therefore, the probability that the woman says her husband does less than his fair share can be calculated as: 1 - (probability of doing more than fair share + probability of doing fair share) = 1 - (0.12 + 0.61) = 0.27.

(b) The probability that a woman believes her husband does at least his fair share is the sum of the probabilities for doing more than his fair share and doing his fair share, which is 0.12 + 0.61 = 0.73.

(a) The sum of the probabilities for different academic ranks must be 1, as it represents all possible outcomes. The sum is calculated as 0.41 + 0.23 + 0.29 + 0.06 + 0.01 = 1.

(b) The probability that a first-year college student was not in the top 20% of their high school class is given by the probability of being in the second, third, fourth, or lowest 20% ranks. Therefore, the probability is 0.23 + 0.29 + 0.06 + 0.01 = 0.59.

(c) The probability that a first-year student was in the top 40% in high school is the sum of the probabilities of being in the top 20% and the second 20%, which is 0.41 + 0.23 = 0.64.