91Ó°ÊÓ

Statistics from the 2009 Major League Baseball season show that there were 157 players who had at least 500 plate appearances. For this group, 42 players had a batting average of 300 or higher, 53 players hit 25 or more home runs, and 14 players had a batting average of .300 or higher and hit 25 or more home runs. Only four players had 200 or more hits (ESPN website, January 10,2010 ). Use the 157 players who had at least 500 plate appearances to answer the following questions. a. What is the probability that a randomly selected player had a batting average of .300 or higher? b. What is the probability that a randomly selected player hit 25 or more home runs? c. Are the events having a batting average of .300 or higher and hitting 25 or more home runs mutually exclusive? d. What is the probability that a randomly selected player had a batting average of .300 or higher or hit 25 or more home runs? e. What is the probability that a randomly selected player had 200 or more hits? Does obtaining 200 or more hits appear to be more difficult than hitting 25 or more home runs? Explain.

Step by step solution

01

Calculate Probability for Batting Average

To find the probability that a randomly selected player had a batting average of .300 or higher, divide the number of players with a batting average of .300 or higher by the total number of players. There are 42 such players out of 157, so the probability is \( \frac{42}{157} \).

02

Calculate Probability for Home Runs

Find the probability that a randomly selected player hit 25 or more home runs by dividing the number of such players by the total number of players. There are 53 players who hit 25 or more home runs out of 157, hence the probability is \( \frac{53}{157} \).

03

Determine if Events Are Mutually Exclusive

Two events are mutually exclusive if they cannot happen at the same time. Since 14 players had both a batting average of .300 or higher and hit 25 or more home runs, these events are not mutually exclusive.

04

Calculate Probability for Either Event

To calculate the probability that a player had a batting average of .300 or higher or hit 25 or more home runs, use the formula: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). Where \(P(A \cap B)\) is the probability of both, calculated as \(\frac{14}{157}\). Thus, \( \frac{42}{157} + \frac{53}{157} - \frac{14}{157} = \frac{81}{157} \).

05

Calculate Probability for 200 or More Hits

Find the probability that a player had 200 or more hits by dividing the number of such players by the total. Four players had 200 or more hits, so the probability is \( \frac{4}{157} \).

06

Compare Home Runs with Hits

Compare the calculated probabilities from Steps 2 and 5. The probability of hitting 25 or more home runs is \( \frac{53}{157} \), which is much higher than \( \frac{4}{157} \) for obtaining 200 or more hits. Thus, obtaining 200 or more hits is more difficult than hitting 25 or more home runs.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Mutually Exclusive Events

In the context of probability, mutually exclusive events are situations where the occurrence of one event prevents the occurrence of another. For example, if you're flipping a coin, landing heads means it cannot be tails, making these events mutually exclusive.
However, in sports statistics, events might overlap. In our baseball case, let's explore the question: Are having a batting average of .300 or higher and hitting 25 or more home runs mutually exclusive? To answer, we check if any player fits both criteria.
According to the original exercise, 14 players had both high batting averages and hit 25+ home runs. This intersection means these events can happen together. When 14 players achieve both, it definitively shows that the events are not mutually exclusive.
This understanding is crucial when calculating probabilities because it affects how we sum them up. Always remember, for mutually exclusive events: if one happens, the other can't. But if events overlap, they are not mutually exclusive.

Probability Calculation

Calculating probability helps us understand how likely something is to occur. To find probability, divide the number of favorable outcomes by the total number of possible outcomes. This gives you a simple fraction, which represents the chance of an event.
For instance, in the baseball scenario, calculating the probability of a player having a .300 or higher batting average requires counting players with this stat and dividing by all the players considered. With 42 such players out of 157 total, the probability is expressed as \( \frac{42}{157} \).

• The probability of hitting 25 or more home runs: 53 players hit this many, so the probability is \( \frac{53}{157} \).
• For the combined event of either a .300+ batting average or 25+ home runs, we use: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). This accounts for double-counting those achieving both.

By understanding this, you can calculate sports-related probabilities and make data-driven decisions or predictions.

Baseball Statistics

Baseball statistics are a fascinating blend of math and sports, giving insight into player performances. Statistics not only inform us about past games but also help forecast future play developments.
Consider how they are applied in our exercise. Specific player achievements, like having a high batting average or hitting numerous home runs, are key aspects.

• Batting Average: Measured by hits divided by at-bats, it's a traditional way to gauge a player's hitting skills.
• Home Runs: Indicates power and ability to score, enhancing a player's offensive value.

In our scenario, noting that only 4 out of 157 players got 200 or more hits informs us about how rare this achievement is compared to hitting 25 or more home runs, which 53 players did.
Thus, analyzing such statistics allows teams to assess player capabilities, strengths, and potential areas for improvement. Remember, understanding stats is crucial for anyone engaged in sports analytics or simply a fan of the game.