Questions: Assume the Poisson distribution applies. Use the given mean to find the indicated probability. Find P(4) when μ=9. P(4)= (Round to the nearest thousandth as needed.)

Solution Steps

We are given a Poisson distribution with a mean \(\lambda = 9\). We need to find the probability of observing exactly 4 events, denoted as \(P(4)\).

The probability of observing \(k\) events in a Poisson distribution is given by the formula: \[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \] where:

• \(\lambda\) is the average number of events,
• \(k\) is the number of events for which the probability is calculated,
• \(e\) is the base of the natural logarithm (approximately 2.71828).

Substitute \(\lambda = 9\) and \(k = 4\) into the formula: \[ P(X = 4) = \frac{9^4 e^{-9}}{4!} \]

Calculate each component:

• \(9^4 = 6561\)
• \(e^{-9} \approx 0.0001234\)
• \(4! = 24\)

Substitute these values back into the formula: \[ P(X = 4) = \frac{6561 \times 0.0001234}{24} \approx 0.0340 \]

Final Answer