91Ó°ÊÓ

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 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 by step solution

01

Identify Given Data

The provided data includes:\( P(\text{Bet}) = 0.17 \) (the probability of an adult betting on professional sports) and \( P(\text{Male}) = 0.484 \) (the probability of an adult being male).

02

Calculate Probability of Male and Betting (Part a)

Since the problem states that betting is independent of gender:\[ P(\text{Male and Bet}) = P(\text{Male}) \times P(\text{Bet}) = 0.484 \times 0.17 = 0.08228 \]

03

Calculation for Male or Betting (Part b)

Use the formula for the union of two events:\[ P(\text{Male or Bet}) = P(\text{Male}) + P(\text{Bet}) - P(\text{Male and Bet}) \]Substitute the values:\[ P(\text{Male or Bet}) = 0.484 + 0.17 - 0.08228 = 0.57172 \]

04

Compare Actual Data (Part c)

\( P(\text{Male and Bet}) \) from the Gallup poll data is given as \( 0.106 \). This is higher than the calculated value 0.08228, indicating the assumption of independence is likely incorrect.

05

Effect on Previous Calculation (Part d)

Recalculate \( P(\text{Male or Bet}) \) using the actual data:\[ P(\text{Male or Bet}) = P(\text{Male}) + P(\text{Bet}) - P(\text{Male and Bet}) \]Substitute the corrected value:\[ P(\text{Male or Bet}) = 0.484 + 0.17 - 0.106 = 0.548 \]This shows a difference from the previous calculation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

independent events

When we talk about independent events in probability, it means the occurrence of one event does not affect the occurrence of another. In the exercise, the assumption that betting and gender are independent allows us to multiply their probabilities straightforwardly.
For example, if the probability of being male is 0.484 and the probability of betting on sports is 0.17, then the probability of being both male and a sports better is calculated as follows:
\[ P(\text{Male and Bet}) = P(\text{Male}) \times P(\text{Bet}) = 0.484 \times 0.17 = 0.08228 \]This is only valid if the two events are truly independent.

union of events

The union of events in probability refers to the scenario where at least one of several specified events occurs. It is calculated using the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]Here, we are finding the probability that an adult is either male or bets on sports. By plugging in the values we have:
\[ P(\text{Male or Bet}) = 0.484 + 0.17 - 0.08228 = 0.57172 \]This formula accounts for any double-counting that happens when considering both independent events.

conditional probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. In this exercise, we initially assume independence, but the data suggests dependency.
Given the updated Gallup poll data that 10.6% of adults are male and bet on sports, we re-evaluate the probability:
\[ P(\text{Male and Bet}) = 0.106 \]Next, we update the union of the two events to see how the incorrect assumption affects results:
\[ P(\text{Male or Bet}) = 0.484 + 0.17 - 0.106 = 0.548 \]This new conditional probability calculation affects the overall understanding of probabilities under non-independence assumptions.

Gallup Poll statistics

A Gallup Poll is a type of public opinion poll that can provide insights into trends and behaviors within a population. In the given exercise, we use data from a Gallup Poll to analyze betting behaviors among American adults.
Initially, the Gallup Poll data showed that 17% of adult Americans bet on professional sports. U.S. Census data indicates that 48.4% of the adult population is male, giving us two probability figures.
The Gallup Poll later indicates 10.6% of adult males bet on sports, suggesting the events are not independent. This helps us understand and correct our assumptions about how events are related in real-world data.
These kinds of statistics are pivotal in making informed decisions based on demographic and behavioral patterns.