Question

A Geiger counter counts the number of alpha particles from radioactive material. Over a long period of time, an average of 7 particles per minute occurs. Assume the arrival of particles at the counter follows a Poisson distribution. Round your answers to 3 decimals.

a. Find the probability that no particle arrives in a particular one minute period.

b. Find the probability that at least one particle arrive in a particular one minute period.

Answer 1

Let X denotes the number of article in a randomly selected one minute period.

X ~ Poisson(7)

The probability mass function of X is

$P(X= x) = \frac{e^{-7}*7^x}{x!},x=0,1,2,....$

a) The probability that no particle arrives in a particular one minute period.

$=P(X= 0) = \frac{e^{-7}*7^0}{0!} = e^{-7} = 0.0009119$

b) The probability that at least one particle arrive in a particular one minute period

$=P(X\geq 1) = 1 - P(X= 0) =1 - \frac{e^{-7}*7^0}{0!} =1 - e^{-7}$

$=1- 0.0009119$

$= 0.999088$