Questions: Suppose that on the average, 4 students enrolled in a small liberal arts college have their automobiles stolen during the semester. What is the probability that exactly 2 students will have their automobiles stolen during the current semester? Round your answer to four decimal places.

Solution Steps

We need to find the probability that exactly \( k = 2 \) students have their automobiles stolen during the semester, given that the average number of students whose cars are stolen is \( \lambda = 4 \).

The probability of observing exactly \( k \) events in a Poisson distribution is given by the formula:

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

Substituting \( \lambda = 4 \) and \( k = 2 \) into the formula, we have:

\[ P(X = 2) = \frac{4^2 e^{-4}}{2!} \]

Calculating each component:

• \( 4^2 = 16 \)
• \( 2! = 2 \)
• \( e^{-4} \approx 0.0183 \) (using the approximate value of \( e \))

Now substituting these values back into the equation:

\[ P(X = 2) = \frac{16 \cdot 0.0183}{2} = \frac{0.2928}{2} = 0.1464 \]

Rounding the result to four decimal places gives:

\[ P(X = 2) \approx 0.1465 \]

Final Answer