Problem 18 - An Introduction to Mathematical Statistics and Its Applications

Question

Assume that the number of hits, \(X\), that a baseball team makes in a nine- inning game has a Poisson distribution. If the probability that a team makes zero hits is \(\frac{1}{3}\), what are their chances of getting two or more hits?

Step-by-Step Solution

Answer

The probability of the baseball team making two or more hits can be computed by subtracting the calculated probabilities of 0 and 1 hit from 1.

Step 1: Understand the Poisson Distribution

The Poisson distribution's probability mass function is represented as \(P(X=k) = \frac{e^{-\lambda} \lambda^{k}}{k!}\), where \(X\) is the number of events (hits in this case), \(\lambda\) is the average rate of occurrence, and \(k\) is the specific number of events we're interested in.

Step 2: Calculate the Poisson Parameter

We are informed that the probability of no hits is \(1/3\), which we can use to obtain \(\lambda\). Plug in \(k = 0\) into the Poisson formula to find \(P(X = 0) = e^{-\lambda}\). Since we know \(P(X = 0) = 1/3\), we can solve for \(\lambda\), yielding \(\lambda = -\ln(1/3)\).

Step 3: Obtain the Probability of One Hit

To obtain the probability of a single hit, plug in \(k = 1\) in the Poisson formula to yield \(P(X = 1) = \lambda e^{-\lambda}\). Substituting for \(\lambda = -\ln(1/3)\), compute the numerical value for \(P(X = 1)\).

Step 4: Find the Probability of Two or More Hits

The requested probability is for two or more hits. The complement rule will be used. Given that the total probability is 1, subtracting the chances of 0 and 1 hits will yield the probability of 2 or more hits. Therefore, \(P(X \geq 2) = 1 - P(X=0) - P(X=1)\). Replace \(P(X=0)\) and \(P(X=1)\) with the corresponding calculated values, and compute the numerical value for \(P(X \geq 2)\).

Key Concepts

Probability Mass Function

The Probability Mass Function (PMF) is a fundamental concept when it comes to understanding the Poisson distribution. It provides a way to determine the probability of observing a certain number of events, such as hits in a baseball game, within a fixed interval. For a Poisson distribution, the PMF is given by the formula:

• \( P(X=k) = \frac{e^{-\lambda} \lambda^{k}}{k!} \)

Here, \( X \) represents the number of hits, \( k \) is the specific instance of hits we are interested in, and \( \lambda \) is the average rate of hits, a key value in characterizing the distribution.

To compute probabilities using this function, you substitute values for \( k \) and \( \lambda \). This allows you to determine the likelihood of exactly \( k \) hits. For instance, in our exercise, you first determine the probability of zero hits, which provides valuable information to calculate other probabilities using this mathematical tool.

Average Rate of Occurrence

In the Poisson distribution, the average rate of occurrence, \( \lambda \), is a crucial parameter. It represents how often we expect the event—in this case, a baseball hit—to happen on average per interval (here a nine-inning game).

To work with a Poisson distribution, knowing \( \lambda \) is essential because it directly influences the probabilities you'll calculate with the PMF. In the exercise, we are given that the probability of zero hits is \( \frac{1}{3} \). This allows us to solve for \( \lambda \) using the formula for zero hits:

• \( P(X=0) = e^{-\lambda} \)

By knowing \( P(X=0) = \frac{1}{3} \), we can determine \( \lambda = -\ln\left(\frac{1}{3}\right) \). Now, \( \lambda \) enables us to find the probability of different numbers of hits during the game.

Understanding \( \lambda \) provides insight into the team's scoring behavior and allows for more accurate probability assessments.

Complement Rule

The Complement Rule is a basic principle in probability that can significantly simplify complex probability calculations. It expresses that the probability of an event occurring is equal to one minus the probability of it not occurring. This is particularly useful when finding the probability of more complicated events such as hitting two or more in baseball.

For the exercise, the complement rule comes into action when determining the probability of making two or more hits. Instead of calculating the direct probability of 2, 3, or more hits individually, you find what you don't need—zero and one hit probabilities:

• The probability of zero hits \( P(X=0) = \frac{1}{3} \).
• The probability of one hit is found by \( P(X=1) = \lambda e^{-\lambda} \). Substitute \( \lambda \) to find its value.

Thus, to find the probability of two or more hits, apply the rule:
\( P(X \geq 2) = 1 - P(X=0) - P(X=1) \).

By calculating these values, you use the complement rule to efficiently reach the desired probability.