Question

2) The Physics Department at a small University averages 6 telephone calls per hour What is the probability that in one hour they receive: a) Exactly 1 call? b) Exactly 2 calls? c) More than two calls? 3) If you had a collection of 1.5x 1020 unstable nuclei, and the probability of any single nuclei decaying in a given minute is 10-20, find: a) The average number of nuclei decaying per minute. b) Is this situation best described by a Binomial distribution, or a Poisson distribution? Explain your answer c) What is the probability that less than 3 nuclei decay in 1 minute?

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Answer 1

Assuming number of telephone calls per hour follow Poisson distribution X with parameter lambda Average telephone calls per hour is 6 Therefore lambda = 6 P(X = x) = e^-lambda lambda^x/x! x = 0, 1,2,3,... a) Probability that in one hour they receive exactly one call is P(X = 1) = e^-lambda lambda^1/1! = e^-6(6)/1 = 6e^-6 = 0.14872513 ~0.015(approx) b) Probability that in one hour they receive exactly 2 calls is P(X = 2) = e^-lambda lambda^2/2! = e^-6(6)^2/2 times 1 = 36e^-6/2 = 18e^-6 = 0.044617539 ~0.045(approx) c) Probability that in one hour they receive more than two calls is P(X > 2) = 1-P(X lessthanorequalto 2) = 1-(P(X = 0) + P(X = 1) + P(X = 2)) = 1-(e^-lambda lambda^0/0! + 6e^-6 + 18e^-6) = 1-(e^-6 + 6e^-6 + 18e^-6) = 1-25e^-6 = 1-25(0.002478752177) = 1-0.06196880442 = 0.93803119 ~0.9380 (approx)