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

Verified step by step guidance

1. Review the probability computed in part (b), which likely involved the overall probability of an adult American betting on professional sports without considering gender.
2. Understand that part (c) introduced additional information, possibly about the probability of betting given the person is male or female, or the proportion of males and females who bet on sports.
3. Recognize that incorporating the gender distribution (48.4% male) and any gender-specific betting probabilities will change the overall probability by weighting the probabilities according to the population proportions.
4. Use the Law of Total Probability to update the overall probability: if \( P(B|M) \) is the probability of betting given male, and \( P(B|F) \) is the probability of betting given female, then the overall probability \( P(B) \) is calculated as \( P(B) = P(B|M) \times P(M) + P(B|F) \times P(F) \), where \( P(M) = 0.484 \) and \( P(F) = 1 - P(M) \).
5. Conclude that the information in part (c) allows for a more precise calculation of the overall betting probability by accounting for gender differences, thus affecting the probability computed in part (b).

Key Concepts

Conditional Probability

Conditional probability measures the likelihood of an event occurring given that another event has already occurred. It adjusts the probability based on new information, such as knowing the gender of a person when calculating the chance they bet on sports.

Independence of Events

Two events are independent if the occurrence of one does not affect the probability of the other. Understanding whether betting behavior depends on gender is crucial to determine if probabilities should be adjusted or remain the same.

Population Proportions and Their Impact on Probability

Population proportions, like the percentage of males in the adult population, influence overall probabilities when combined with subgroup behaviors. Incorporating demographic data helps refine probability estimates by weighting according to group sizes.