Problem 38 - 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- at least one of the two friends is able to get a hit their first time at bat?

Step-by-Step Solution

Answer

0.58

Step 1: Understand the Problem

We need to find the probability that at least one of Albert or Paul gets a hit in their first time at bat. This is a classic probability problem involving 'at least one' scenarios.

Step 2: Calculate Individual Probabilities

Albert's probability of getting a hit is 0.4. Therefore, the probability of Albert not getting a hit is \(1 - 0.4 = 0.6\). Similarly, Paul's probability of getting a hit is 0.3, so the probability of Paul not getting a hit is \(1 - 0.3 = 0.7\).

Step 3: Calculate Combined Probability of Both Missing

The probability that both Albert and Paul do not get hits is the product of their individual probabilities of not getting hits. Thus, it is \(0.6 \times 0.7 = 0.42\).

Step 4: Calculate Probability of At Least One Hit

To find the probability that at least one gets a hit, subtract the probability of both missing from 1. Therefore, the probability is \(1 - 0.42 = 0.58\).

Key Concepts

Batting Average

Batting average is a useful statistic in baseball that depicts a player's ability to hit the ball. It is calculated by dividing the number of hits by the number of at bats. This statistic is expressed as a decimal and often shown as a percentage. In our exercise, Albert has a batting average of 0.4 or 40%, implying he gets a hit 40% of the times he swings. Paul, on the other hand, has a batting average of 0.3 or 30%.
These averages help predict the likelihood of a specific outcome, such as getting a hit, which is essential when calculating probabilities in probability theory.
When working with batting averages in probability problems, it's important to translate these percentages into decimal form to perform calculations correctly.

Independent Events

In probability, independent events mean the outcome of one event does not affect the outcome of another. For example, when Albert and Paul step up to bat, Albert's success at getting a hit does not influence whether Paul gets a hit. Each event is his own separate chance.

The mathematical characteristic of independent events is illustrated by the rule: the probability of both events occurring together is the product of the probabilities of each event occurring individually. This is why in our example, the probability of both Albert and Paul missing their hits is calculated as the product of their separate probabilities of failure, i.e., 0.6 (for Albert) multiplied by 0.7 (for Paul).
This concept is crucial for many probability scenarios beyond batting averages, including things like rolling dice or flipping coins.

Complementary Probability

Complementary probability refers to the idea that the sum of the probabilities of all possible outcomes of an event equals 1. For any event, the probability that it will not occur is the complement of the probability that it will occur.

In the context of our exercise, if Albert has a 0.4 probability of hitting, the complementary probability that he will not hit is 1 - 0.4 = 0.6. Similarly, if Paul's batting probability is 0.3, his probability of not hitting is 1 - 0.3 = 0.7.

• This complements the calculations needed for finding other probabilities, as seen in the problem where we use the complementary probabilities to determine the likelihood of neither batter hitting.
• Understanding complementary probability is essential for calculating scenarios where you want to understand the chances of the opposite or combined events.

At Least One Scenario

The phrase 'at least one scenario' in probability indicates the event when one or more specific outcomes occur. Rather than calculating each instance where only one succeeds, it's often easier to calculate the scenario where none succeed and subtract from total probability.

In our problem, rather than figuring out all combinations where Albert or Paul might hit, we calculate the probability that neither scores a hit and subtract this from 1. This gives the probability that at least one of them hits the ball. Mathematically, for the probability of at least one hitting:

• Calculate the scenario where none hit: 0.6 (Albert misses) * 0.7 (Paul misses) = 0.42.
• The probability of at least one hit is then 1 - 0.42 = 0.58.

This approach streamlines calculation and illustrates how complementary events connect with practical scenarios.