91Ó°ÊÓ

The Wall Street Journal regularly publishes an article entitled "The Count." In one article, The Count looked at 1000 randomly selected home runs in Major League Baseball. (a) Of the 1000 homeruns, it was found that 85 were caught by fans. What is the probability that a randomly selected homerun is caught by a fan? (b) Of the 1000 homeruns, it was found that 296 were dropped when a fan had a legitimate play on the ball. What is the probability that a randomly selected homerun is dropped? (c) Of the 85 caught balls, it was determined that 34 were barehanded catches, 49 were caught with a glove, and two were caught in a hat. What is the probability a randomly selected caught ball was caught in a hat? Interpret this probability. (d) Of the 296 dropped balls, it was determined that 234 were barehanded attempts, 54 were dropped with a glove, and eight were dropped with a failed hat attempt. What is the probability a randomly selected dropped ball was a failed hat attempt? Interpret this probability.

Step by step solution

01

Probability a Homerun is Caught by a Fan

To find the probability that a randomly selected homerun is caught by a fan, divide the number of caught homeruns by the total number of homeruns.Let the total number of homeruns be 1000 and the number caught by fans be 85.Using the formula for probability: \[P(\text{caught by a fan}) = \frac{\text{number of caught homeruns}}{\text{total number of homeruns}}\]\[P(\text{caught by a fan}) = \frac{85}{1000} = 0.085\]

02

Probability a Homerun is Dropped

To find the probability that a randomly selected homerun is dropped, divide the number of dropped homeruns by the total number of homeruns.Let the total number of homeruns be 1000 and the number dropped be 296.Using the formula for probability: \[P(\text{dropped}) = \frac{\text{number of dropped homeruns}}{\text{total number of homeruns}}\]\[P(\text{dropped}) = \frac{296}{1000} = 0.296\]

03

Probability a Caught Ball was Caught in a Hat

To find the probability that a randomly selected caught ball was caught in a hat, divide the number of balls caught in a hat by the number of caught balls.Let the total number of caught balls be 85 and the number caught in a hat be 2.Using the formula for probability: \[P(\text{caught in a hat}) = \frac{\text{number of balls caught in a hat}}{\text{total number of caught balls}}\]\[P(\text{caught in a hat}) = \frac{2}{85} \approx 0.0235\]This means that there is approximately a 2.35% chance that a randomly selected caught ball was caught in a hat.

04

Probability a Dropped Ball was a Failed Hat Attempt

To find the probability that a randomly selected dropped ball was a failed hat attempt, divide the number of failed hat attempts by the number of dropped balls.Let the total number of dropped balls be 296 and the number of failed hat attempts be 8.Using the formula for probability: \[P(\text{failed hat attempt}) = \frac{\text{number of failed hat attempts}}{\text{total number of dropped balls}}\]\[P(\text{failed hat attempt}) = \frac{8}{296} \approx 0.0270\]This means that there is approximately a 2.70% chance that a randomly selected dropped ball was a failed hat attempt.

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Key Concepts

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

Probability

Probability is the measure of how likely an event is to happen. In mathematical terms, it is the ratio of the favorable outcomes to the total number of possible outcomes. The probability of an event A is denoted as \( P(A) \), and is calculated using the formula: \[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]
In the context of sports statistics, probability helps us understand the likelihood of different events, such as a home run being caught by a fan or dropped.

Sports Statistics

Sports statistics involve collecting, analyzing, and interpreting data related to athletic events. This data can include anything from player performance metrics to game outcomes and fan interactions. Analyzing sports statistics can help teams improve strategies, enhance player performance, and predict future outcomes.
In the given exercise, statistics are used to track how often home runs are caught or dropped by fans, providing valuable insights into fan interaction during the games.

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. It helps in understanding how the probability of one event changes in the context of another. The conditional probability of event A given event B is represented by \(P(A|B)\) and is calculated using the formula: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
For example, in the exercise, the probability of a ball being caught in a hat is conditional on it being caught by a fan in the first place.

Data Interpretation

Data interpretation involves analyzing data to draw meaningful conclusions. This process is essential in transforming raw data into useful information.

• First, identify the total sample size and the number of occurrences of the specific event.
• Next, apply the appropriate probability formulas to calculate probabilities.
• Finally, interpret the probability values to understand what they represent in context.
  Practicing these steps in various contexts, like sports statistics, can enhance your ability to draw accurate and meaningful conclusions from data.