Question

4. Geologists estimate the time since the most recent cooling of a mineral by counting the number of uranium fission tracks on the surface of the mineral. A certain mineral specimen is of such an age that there should be an average of 6 tracks per cm2 of surface area. Assume the number of tracks in an area follows a Poisson distribution. Let X represent the number of tracks counted in 1 cm2 of surface area.

a)Find P(X = 7).

b)Find P(X ≥ 3).

c)Find P(2 < X < 7).

d)Find μX.

e)Find σX

Answer 1

In this case we use the Poisson distribution because we are talking about the occurrence of an event (number of tracks) over a specified interval (in this case an area interval).

The probability of the event occurring x times over an interval is:

P(x) = nˣ × e⁻ⁿ ÷ x!

where n is the mean.

a) P(7) = 6⁷ × e⁻⁶ ÷ 7! = 0.1376

b) P(x ≥ 3) = 1 - P(x < 3) = 1 - P(2) - P(1) - P(0)

P(2) = 6² × e⁻⁶ ÷ 2! = 0.0446

P(1) = 6¹ × e⁻⁶ ÷ 1! = 0.0149

P(0) = 6⁰ × e⁻⁶ ÷ 0! = 0.0025

P(x ≥ 3) = 0.9380

c) P(2 < x < 7) = P(3) + P(4) + P(5) + P(6) = 0.0892 + 0.1339 + 0.1606 + 0.1606 = 0.5443

d) The mean is going to be 6.

e) The standard deviation is √n = √6 = 2.4