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)

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

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

$$\approx0.8648$$