Question

In a baseball game, Tommy gets a hit \(30 \%\) of the time when facing this pitcher. Joey gets a hit \(25 \%\) of the time. They are both coming up to bat this inning. a. What is the probability that Joey or Tommy will get a hit? b. What is the probability that neither player gets a hit? c. What is the probability that they both get a hit?

Short answer

a. 0.475 b. 0.525 c. 0.075

Step-by-step solution

Understanding Individual Probabilities

Tommy's probability of getting a hit is 0.30 and Joey's probability of getting a hit is 0.25.

Calculate Probability of Either Joey or Tommy Getting a Hit

Use the formula for the probability of either event happening: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \) where \( P(A \cap B) \) is the probability that both happen. \( P(T) = 0.30 \), \( P(J) = 0.25 \). Substitute these into the formula: \( 0.30 + 0.25 - (0.30 \times 0.25) = 0.30 + 0.25 - 0.075 = 0.475 \). The probability of either Joey or Tommy getting a hit is 0.475.

Calculate Probability that Neither Player Gets a Hit

This is the complement of at least one of them getting a hit. Since the probability of either getting a hit is 0.475, the probability that neither gets a hit is \(1 - 0.475 = 0.525\).

Calculate Probability that Both Players Get a Hit

The probability that both get a hit is the product of their individual probabilities: \( P(T \cap J) = P(T) \times P(J) = 0.30 \times 0.25 = 0.075 \).

Key concepts

Complement Rule

The complement rule is a useful concept in probability. It allows you to calculate the probability of an event not happening by subtracting the probability of the event happening from 1. This makes it easy to find the probability of the opposite scenario. In the baseball example, we are interested in finding out the probability that neither Tommy nor Joey gets a hit.
To use the complement rule here, we first found the probability that either Tommy or Joey does get a hit, which is 0.475. The complement of this scenario, meaning neither of them get a hit, is simply:

• Subtracting 0.475 (probability of either getting a hit) from 1, gives us 0.525.

This tells us that there is a 52.5% chance that neither player makes a hit when facing the pitcher.

Intersection Probability

Intersection probability deals with the likelihood of two events happening at the same time. In other words, it calculates the probability that both events occur simultaneously. For instance, in our baseball example, this would mean both Tommy and Joey hitting the ball in a single inning.
To determine this, you multiply the probabilities of each individual event happening:

• Tommy's probability of hitting is 0.30.
• Joey's probability of hitting is 0.25.

So, the probability of both of them hitting is given by the formula:\[P(T \cap J) = P(T) \times P(J) = 0.30 \times 0.25 = 0.075\]This calculation shows us that there is a 7.5% chance that both Tommy and Joey will get a hit when they are up against the pitcher this inning.

Union Probability

Union probability is about finding the chance of either one or another event happening, or both. In probability terms, the word 'union' signifies the occurrence of any one, or more, of several events. In our case, we want to find the probability that either Tommy or Joey, or both, get a hit.
We use the union probability formula here, which is:

• \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]

Where

• \(P(A)\) is the probability of Tommy getting a hit (0.30),
• \(P(B)\) is the probability of Joey getting a hit (0.25),
• \(P(A \cap B)\) is the probability that both get a hit (0.075).

By substituting these values in, we calculated:\[0.30 + 0.25 - 0.075 = 0.475\]This means there's a 47.5% chance that either Tommy or Joey gets a hit in this inning. This formula helps avoid over-counting the situation where both could happen simultaneously.