Answer to Question #90984 in Statistics and Probability for Usman Ahimed

Question

suppose that flaws in plywood occur at random with an average of one flaw per 50 square feet. what is the probability a 4 foot * 8 foot sheet will have no flaws,At most one flaws....

Answer 1

To get a solution assume that the number of flaws per unit area is Poisson distributed.

Let X be the Poisson random variable with the mean equal to $E(X)=\lambda \dfrac{S}{S_0}.$

Formula Poisson probability

$$P(X=k)={(\lambda{S \over S_0})^k\cdot e^{-(\lambda{S \over S_0})} \over k!}$$

Given that

$$\lambda=1, S_0=50 ft^2, S=4ft\cdot 8ft=32ft^2$$

Then

$$\lambda{S \over S_0}=1\cdot {32ft^2 \over 50ft^2}=0.64$$

a)

$$P(X=0)={0.64^0\cdot e^{-0.64} \over 0!}=e^{-0.64}\approx0.5273$$

b)

$$P(X\leq1)=P(X=0)+P(X=1)=$$

$${0.64^0\cdot e^{-0.64} \over 0!}+{0.64^1\cdot e^{-0.64} \over 1!}=e^{-0.64}+0.64e^{-0.64}=1.64e^{-0.64}\approx$$

$$\approx0.8648$$