Answer to Question #304635 in Statistics and Probability for lak

Question

A certain kind of sheet metal has, on average, 9 defects per 14 square feet.

Assuming a Poisson distribution, find the probability that a 25 square foot metal sheet has at least 14 defects. Round your answer to four decimals.

Answer 1

Let $X=$ the number of defects: $X\sim Po(\lambda).$

$$P(X=k)=\dfrac{e^{-\lambda}(\lambda)^k}{k!}$$

Given $\lambda=\dfrac{25}{14}\cdot 9\approx16.07143$

$$P(X\ge14)=1-P(X<14)$$

$$=1-P(X=0)-P(X=1)-P(X=2)$$

$$-P(X=3)-P(X=4)-P(X=5)$$

$$-P(X=6)-P(X=7)-P(X=8)$$

$$-P(X=9)-P(X=10)-P(X=11)$$

$$-P(X=12)-P(X=13)\approx0.3609$$