Question

If the number of flaws on a piece of plywood occurs randomly and follows the Poisson distribution with a mean of 1 per 5m^2

.

(a) Find the probability that a piece of randomly chosen plywood will contain no flaws.

(b) The plywood is sold in pieces of size 3m^2. Find the probability that the plywood will contain no flaws.

Answer 1

$\text{Poisson's formula:}$

$P(X=m)=\frac{\lambda^m{e^{-\lambda}}}{m!}$

$a)\text{ lets a piece of plywood has an area } S$

$\text{then }\lambda=\frac{S}{5}$

$P(0) -\text{the probability that the plywood does not contain flaws}$

$P(0)=\frac{\lambda^0{e^{-\lambda}}}{0!}=e^{-\lambda}=e^{-\frac{S}{5}}(1)$

$b)\text{we use the formula (1)}$

$P(0)= e^{-\frac{3}{6}}\approx0.61$

Answer:

$a)P(0)=e^{-\frac{S}{5}},\text{where S area of a piece of plywood}\newline b)P(0)=0.6$