Question

Molecules of a rare gas occur at an average rate of 3 per cubic metre of air and follow a poisson distribution.
i) Find the probability that 3 cubic metre of air contain exactly 4 of the molecules
ii) The probability that at least one molecule is found is 0.99. Find how much air is taken as a sample.

Answer 1

i. Poisson(3×3)

$P(4)=\frac{9^4e^{-9}}{4!}=0.03374$

ii 1-P(0)=0.99

$1-\frac{(3x)^0e^{-3x}}{0!}=0.99$

$0.01=e^{-3x}$

$\ln0.01=-3x$

$x=\frac{\ln0.01}{-3}=1.5351$

1.5351 cubic metres of air is taken.