Problem 37 - Algebra 2

Question

BASEBALL For Exercises \(36-38\) , use the following information. Albert and Paul are on the school baseball team. Albert has a batting average of \(.4,\) and Paul has a batting average of 3 . That means that Albert gets a hit 40\(\%\) of his at bats and Paul gets a hit 30\(\%\) of his times at bat. What is the probability that- neither Albert nor Paul is able to get a hit their first time at bat?

Step-by-Step Solution

Answer

The probability is 0.42.

Step 1: Understand the Problem

We need to find the probability that neither Albert nor Paul gets a hit in their first at bat. This involves calculating the probability that Albert does not get a hit and Paul does not get a hit.

Step 2: Calculate Probability of No Hit for Albert

Albert's batting average is 0.4, which means he gets a hit 40% of the time. The probability that Albert does not get a hit is the complement of getting a hit, which is 1 - 0.4 = 0.6 or 60%.

Step 3: Calculate Probability of No Hit for Paul

Paul's batting average is 0.3, meaning he gets a hit 30% of the time. The probability that Paul does not get a hit is 1 - 0.3 = 0.7 or 70%.

Step 4: Calculate Combined Probability

The probability that neither Albert nor Paul gets a hit in their first at bat is the product of the individual probabilities. Multiply the probability of Albert not getting a hit by the probability of Paul not getting a hit: 0.6 * 0.7 = 0.42.

Key Concepts

Understanding Complementary Events

In probability, complementary events are a fundamental concept. A complementary event refers to the situation where an event either happens or it doesn’t happen — it's an either-or situation. If the probability of an event happening is known, the probability that the event does not happen can be easily calculated.

To find the probability of a complementary event, you subtract the probability of the event from one. Mathematically, when an event has a probability of occurring, denoted as \(P(A)\), the probability that the event will not occur is \(1 - P(A)\). This formula is useful whenever dealing with probabilities, as seen in our exercise example where the probabilities that Albert and Paul do not get a hit are calculated as complementary to their batting averages.

This simple subtraction is important because it allows us to work with the probability of an event's complementary which often simplifies tackling probability questions. In this exercise, understanding complementary events was key to determining the chances of both players missing their hits.

Deciphering Batting Averages in Probability

A batting average is a statistic in baseball that gives insight into a player's hitting skills by indicating how often a player gets a hit. It represents the fraction of times a player hits compared to the total number of their at-bats. Let's break it down to understand how it fits into probability.

If a player like Albert has a batting average of 0.4, this means Albert gets a hit 40% of the time when he bats. This data can be directly translated to determine probabilities in relevant problems. To find out the probability of the opposite, or the complementary event (not getting a hit), simply subtract the batting average from one.

• Albert's average of 0.4 gives a probability of getting no hit as \(1 - 0.4 = 0.6\)
• Paul's average of 0.3 gives a probability of no hit as \(1 - 0.3 = 0.7\)

Batting averages are a practical example of using decimal or proportion forms directly as probabilities in problem-solving, offering a straightforward way to handle statistical analysis in sports contexts.

Applying the Multiplication Rule for Probabilities

The multiplication rule is a critical concept in probability, used to find the joint probability of two independent events happening together. Understanding this rule is essential when confronted with questions asking for the probability of multiple events occuring simultaneously.

When two events, say Event A and Event B, are independent, the probability of both occurring is the product of their individual probabilities. In mathematical terms, if \(P(A)\) represents the probability of event A, and \(P(B)\) the probability of event B, the combined probability is \(P(A \text{ and } B) = P(A) \times P(B)\).

Returning to our exercise scenario, the probability that neither Albert nor Paul gets a hit can be calculated using this rule:

• Probability of no hit from Albert: 0.6
• Probability of no hit from Paul: 0.7
• Combined probability using multiplication rule: \(0.6 \times 0.7 = 0.42\)

This product gives a 42% chance that neither player scores a hit, perfectly illustrating the application of the multiplication rule for independent events. Once mastered, this rule simplifies the process of solving complex probability problems with multiple independent events.