Consider a situation in which a researcher is carbon dating a sample of charcoal

Question

Consider a situation in which a researcher is carbon dating a sample of charcoal from an archaeological site and using a Geiger counter to detect decay events. Radioactive decay is a Poisson process and with the sample the researcher is using she determines that the probabilities of detecting 3 or 4 decay events in a minute are 13.25% and 5.12% respectively. Use a value of e=2.7 for your calculations if needed and provide all answers to the nearest 0.001. With the sample being examined, what is the mean number of decay events per minute? What is the probability that the researcher detects 2 decay events in a given minute? What is the probability that the researcher detects multiple decay events in a given minute?

Explanation / Answer

Solution

Back-up Theory

For a Poisson process with parameter , P(k events) = e- k/(k!)

Now, to work out the solution,

Part (a)

For a Poisson process with parameter , the mean = .

Converting the given probabilities into Poisson probability form, if X = number of decay events in a minute, then

P(X = 3) = 0.1325 or e- 3/(3!) = 0.1325 or e- 3 = 0.795 ……………. (1)

P(X = 4) = 0.0512 or e- 4/(4!) = 0.0512 or e- 4 = 1.2288 ……….…. .(2)

(2)/(1): = 1.2288/0.795 = 1.5457 ~ 1.546 ANSWER

Part (b)

P(two decay events in a minute) = P(X = 2) = e- 1.546(1.546)2/(2!) = 0.2547 ~ 0.255 ANSWER

Part (c)(3)

P(multiple decay events in a minute) = P(X = 1,2, 3,…..) = 1 – P(X = 0) = 1 - e- 1.546

= 1 – 0.2131 = 0.7869 ~ 0.787 ANSWER

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