Question

assume the poisson distribution applies. use the given mean to find the indicated probability. find p(6) when μ = 7. p(6) = (round to the nearest thousandth as needed.)

Explanation:

Step1: Recall Poisson Probability Formula

The formula for the Poisson probability is \( P(x) = \frac{\mu^x e^{-\mu}}{x!} \), where \( \mu \) is the mean, \( x \) is the number of occurrences, and \( e \) is the base of the natural logarithm (approximately 2.71828).

Step2: Identify Values

Here, \( \mu = 7 \), \( x = 6 \), \( e \approx 2.71828 \).

Step3: Substitute into Formula

First, calculate \( \mu^x = 7^6 = 117649 \). Then, calculate \( e^{-\mu} = e^{-7} \approx 0.00091188 \). Next, calculate \( x! = 6! = 720 \). Now, substitute these into the formula: \( P(6) = \frac{7^6 \times e^{-7}}{6!} = \frac{117649 \times 0.00091188}{720} \).

Step4: Perform Calculations

First, multiply the numerator: \( 117649 \times 0.00091188 \approx 107.204 \). Then, divide by 720: \( \frac{107.204}{720} \approx 0.149 \).

Answer:

\( 0.149 \)