Suppose the counts recorded by a Geiger counter follow a Poi

Suppose the counts recorded by a Geiger counter follow a Poisson process with an average of two counts per minute. Round the answers to 3 decimal places.

a) What is the probability that there are no counts in a 30-second interval?

b) What is the probability that the first count occurs in less than 7 seconds?

c) What is the probability that the first count occurs between 1 and 2 minutes after start-up?

Solution

mean per one minute = 2 count /minute

mean per 30 second = 1 count/ 30 second.

P(x=r) = e^(-mean) * mean^r /r!

P(x=0) = e(-1) * 1^0/0! = 0.3678

b.

counter less than 7 second,

so count in first 6 seconds,

probability of count in a second = 1/30 count/second.

p = 1/30

q = 29/30

P(x<7) = P(x=1) + P(x=2) + P(x=3) +....+P(x=6)

P(x<7) = p + pq + pq^2 + pq^3 + pq^4 + pq^5

P(x<7) = p(1+q+q^2+q^3+q^4+q^5)

P(x<7) = p * [ 1-q^6] / [1-q]

P(x<7) = 1 - q^6

for universal,

P(x<r) = 1 - q^(r-1)...................(1)

P(x<7) = 1 - (29/30) ^ 6 = 0.1840

c.

probability between 1 minute to 2 minute,

P(59<x<121) = P(x<121) - P(x<59)

by taking value from equation 1,

P(59<x<121) = [ 1 - q^(120) ] - [ 1-q^58]

P(59<x<121) = [ 1 - (29/30)^120 ] - [ 1 - (29/30)^58 ]

P(59<x<121) = 0.1229