Problem 32E

Betting on Sports According to a Gallup Poll, about 17% of adult Americans bet on professional sports. Census data indicate that 48.4% of the adult population in the United States is male.

(a) Assuming that betting is independent of gender, compute the probability that an American adult selected at random is a male and bets on professional sports.

(b) Using the result in part (a), compute the probability that an American adult selected at random is male or bets on professional sports.

(c) The Gallup poll data indicated that 10.6% of adults in the United States are males and bet on professional sports. What does this indicate about the assumption in part (a)?

(d) How will the information in part (c) affect the probability you computed in part (b)?

Step 1 of 3:

About 17% of adult Americans bet on professional sports.

Census data indicate that 48.4% of the adult population in the United States is male.

We have to find the following.

The probability that an American adult selected at random is male and bets on professional sports, by assuming that betting is independent of gender.The probability that an American adult selected at random is male or bets on professional sports by using part (a).The poll data indicated that 10.6% of adults in the United States are males and bet on professional sports, we have to describe what does this indicate about the assumption in part (a).We have to describe how will the information in part (c) affect the probability that computed in part (b).Step 2 of 3:

Let us define the events:

A: {Americans bet on professional sports}

B: {Adult population in the United States is male}

From the given information:

P (A) = 0.17

P (B) = 0.484

(a)

Here betting is independent of gender.

Two events are said to be independent if P (AB) = P (A)P (B).

P (The selected adult is a male and bets on professional sports) = P (AB)

= P (A)P (B)

= 0.17 0.484

= 0.082

Therefore, the probability that an American adult selected at random is male and bets on professional sports, by assuming that betting is independent of gender is 0.082.

(b)

By the addition rule of probability:

P (AB) = P (A)+P (B)-P (AB)

P (The selected adult is a male or bets on professional sports) = P(AB)

= P (A)+P (B)-P(A)

[ from part (a)]

= 0.17+0.484-(0.082)

= 0.572

Therefore, the probability that an American adult selected at random is male or bets on professional sports is 0.572.