Radioactive atoms are unstable because they have too much energy. When they rele

Question

Radioactive atoms are unstable because they have too much energy. When they release their extra energy, they are said to decay. When studying cesium-137, it is found that during the course of decay over 365 days, 1,000,000 radioactive atoms are reduced to 977, 287 radioactive atoms. a. Find the mean number of radioactive atoms lost through decay in a day. a. 0.2347 b. 134,000 c. 0.8745 d. 22, 713 Find the probability that on a given day, 50 radioactive atoms decayed. a. 0.0155 b. 0.2225 c. 0.9999 d. 0.6543

Explanation / Answer

22713 atoms have decayed over the year

22713 / 365 = 62.2274 atoms decay per day.

Now we will use the Poisson distribution to finish the question.


In general, if X has the Poisson distribution with a rate of w events in t time units then:

X ~ Poisson(wt)
P(X = x) = (wt)^x * exp(-wt) / x! for x = 0, 1, 2, 3, 4, ...
P(X = x) = 0 otherwise

the mean of the Poisson distribution is the parameter, wt
the variance of the Poisson distribution is the parameter, wt


Here we have X = number of atoms decaying in one day and X ~ Poisson(wt = 62.2274)

Find P(X = 50)

= 62.2274^50 exp(-62.2274) / 50!
= 0.01552356

the probability there will be exactly 50 decays in one day is 0.01552356

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