Question

The average number of defects per 20 m2 area of a floor mat produced by a local company is two. A room has an area of 100 m2 and uses the same floor mat as described above.
i) Find the probability that there are at most two defects in a randomly selected 20 m2 of the floor mat.
ii) Find the probability that the floor mat contains at least three defects in the room.

Answer 1

a) We need to use a Poisson distribution with $\lambda=2$

$$P(k\leq 2 )=P(k=0)+P(k=1)+P(k=2)$$

$$P(k\leq 2 )=2^0\frac{e^{-2}}{0!}+2^1\frac{e^{-2}}{1!}+2^2\frac{e^{-2}}{2!}$$

$$P(k\leq 2 )=e^{-2}(1+2+2)=5e^{-2}=0.6767$$

b) We need to use a Poisson distribution with $\lambda'=2\frac{100}{20}=2(5)=10$

$$P(k\geq 3 )=1-P(k\leq 2 )$$

$$P(k\leq 2 )=10^0\frac{e^{-10}}{0!}+10^1\frac{e^{-10}}{1!}+10^2\frac{e^{-10}}{2!}$$

$$P(k\leq 2 )=e^{-10}(1+10+50)=61e^{-10}=0.0028$$

$$P(k\geq 3 )=1-0.0028=0.9972$$