According to a Gallup Poll, about 17 % of adult Americans...

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 Galluppolldata 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)?

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 Galluppolldata 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)?

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Key Concepts

Independence in Probability

This concept refers to the condition where the occurrence of one event does not affect the occurrence of another. Under independence, the probability of both events occurring is found by multiplying their individual probabilities, which simplifies many calculations in probability theory.

Multiplication Rule

The multiplication rule states that if two events are independent, the probability that both occur is the product of their separate probabilities. This rule is essential when dealing with joint probabilities, and it assumes that one event does not influence the other.

Union of Events and the Inclusion?Exclusion Principle

The inclusion-exclusion principle is used to calculate the probability that at least one of two events occurs. It is computed by adding the probabilities of the individual events and then subtracting the probability of their intersection, correcting for any overlap between the events.

Dependence Between Events

This concept refers to the scenario where the occurrence of one event affects the probability of another event. When events are dependent, the product of the individual probabilities does not equal the probability of their intersection, which can lead to different results than those predicted under an independence assumption.

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