Problem 36 - 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- both Albert and Paul are able to get hits their first time at bat?

Step-by-Step Solution

Answer

The probability is 0.12.

Step 1: Identify the Probabilities

First, determine the probability that each player gets a hit at their first at bat. For Albert, it is given as 40%, which can be expressed as a probability: \( P(A) = 0.4 \). For Paul, the probability is 30%, so \( P(P) = 0.3 \).

Step 2: Calculate Joint Probability

Next, calculate the probability that both Albert and Paul get hits. Since these events are independent, multiply the individual probabilities: \( P(A \cap P) = P(A) \times P(P) = 0.4 \times 0.3 \).

Step 3: Compute the Final Answer

Complete the multiplication from the previous step: \( P(A \cap P) = 0.4 \times 0.3 = 0.12 \). Therefore, the probability that both Albert and Paul get hits their first time at bat is 0.12.

Key Concepts

Independent Events

In probability, independent events are those whose outcomes do not influence each other. For example, when you flip a coin, the result of one flip doesn't affect the result of the next flip. Similarly, in our baseball scenario, Albert's ability to get a hit does not affect Paul's chances, and vice versa.
The crucial point is that knowing the outcome of one event gives you no information about the other. If events are independent, their joint probability can be determined by multiplying the probability of each event. This is because their outcome is unrelated, as if they exist in their own worlds.

• Albert's success rate at batting doesn't change based on Paul's performance.
• Paul's batting average remains constant, whether or not Albert gets a hit.

Understanding independent events is key because it simplifies the calculation of joint probabilities, especially in scenarios with multiple actors or actions, like sports events.

Batting Average

A batting average is a statistic in baseball that offers an insight into a player's performance based on their number of at-bats. It is calculated by dividing the number of hits a player makes by their total number of at-bats. In simpler terms, it indicates the likelihood of a player getting a hit. For example, if Albert's batting average is 0.4, it means he successfully makes a hit in 40% of his at-bats.

• Albert's batting average: 0.4 reflects a 40% success rate.
• Paul's batting average: 0.3 reflects a 30% success rate.

This statistic is essential for coaches and players as it provides a clear picture of a player's hitting efficiency. Higher batting averages suggest better performance at the plate, making those players crucial in forming batting strategies for baseball games.

Joint Probability

Joint probability refers to the likelihood of two independent events both occurring at the same time. For instance, when considering Albert and Paul's batting scenarios, we are interested in the probability of both players getting a hit in the same at-bat attempt.
To calculate this, you multiply their individual probabilities together since the hits are independent events. The formula is given by:
\[P(A \cap P) = P(A) \times P(P)\]
Plugging in the numbers from the example, we get:
\[P(A \cap P) = 0.4 \times 0.3 = 0.12\]
This tells us there's a 12% chance that both Albert and Paul will get a hit on their first try. Calculating joint probability is crucial in various scenarios, not just sports, as it helps in predicting combined outcomes in fields like finance, statistics, and everyday decision-making.