Question

In a poll, each of a certain number of American adults was asked the question "If you had to choose, which ONE of the following sports would you say is your favorite?" Of the survey participants, 31% chose pro football as their favorite sport. The report also included the statement "Adults with household incomes of

$75,000 – < $100,000 (48%)

are especially likely to name pro football as their favorite sport, while love of this particular game is especially low among those in $100,000+ households (22%)."

Suppose that the percentages from this poll are representative of American adults in general. Consider the following events.

F	=	event that a randomly selected American adult names pro football as his or her favorite sport
L	=	event that a randomly selected American has a household income of $75,000 – < $100,000
H	=	event that a randomly selected American has a household income of $100,000+

(a)

Use the given information to estimate the following probabilities.

(i)

P(F)

(ii)

P(F|L)

(iii)

P(F|H)

(b)

Are the events F and L mutually exclusive? Justify your answer.

Yes, F and L are mutually exclusive because one can name pro football as a favorite sport and have a household income of $75,000 – < $100,000.Yes, F and L are mutually exclusive because one cannot name pro football as a favorite sport and have a household income of $75,000 – < $100,000.    No, F and L are not mutually exclusive because one can name pro football as a favorite sport and have a household income of $75,000 – < $100,000.No, F and L are not mutually exclusive because one cannot name pro football as a favorite sport and have a household income of $75,000 – < $100,000.

(c)

Are the events H and L mutually exclusive? Justify your answer.

Yes, H and L are mutually exclusive because a household income cannot be both $75,000 – < $100,000 and $100,000+.Yes, H and L are mutually exclusive because a household income can be both $75,000 – < $100,000 and $100,000+.    No, H and L are not mutually exclusive because a household income cannot be both $75,000 – < $100,000 and $100,000+.No, H and L are not mutually exclusive because a household income can be both $75,000 – < $100,000 and $100,000+.

(d)

Are the events F and H independent? Justify your answer.

Yes, F and H are independent because P(F) ≠ P(F|H).Yes, F and H are independent because P(F) = P(F|H).    No, F and H are not independent because P(F) ≠ P(F|H).No, F and H are not independent because P(F) = P(F|H).

Answer 1

Given:

P(F) = 0.31
P(L) = 0.48
P(H) = 0.22

a)

i. P(F) = 0.31

ii. Using the conditional probability law,

P(F|L) =P(F and L)/P(L)

Since the F and L are independent events, P(F and L) = P(F)P(L)

hence,

P(F|L) = P(F) = 0.31

iii.

similarly, F and h are independent events

P(F|H) = P(F) = 0.31

b)   

Answer: No, F and L are not mutually exclusive because one can name pro football as a favorite sport and have a household income of $75,000 – < $100,000.

Explanation: Since, both the event F and L can coincide, these are not mutually exclusive events.

c)

Answer: Yes, H and L are mutually exclusive because a household income cannot be both $75,000 – < $100,000 and $100,000+.

Explanation: Since events, H and L are the two different household income groups, these are mutually exclusive events.

d)

Answer: Yes, F and H are independent because P(F) = P(F|H)

Explanation: Since the events F such that a randomly selected American adult names pro football and the event H such that a randomly selected American adult has a household income of $100,000+ doesn't depends on each other, these are independent events.