In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.

Radioactive Decay 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, a nuclear engineer found that over 365 days, 1,000,000 radioactive atoms decayed to 977,287 radioactive atoms; therefore 22,713 atoms decayed during 365 days.

a. Find the mean number of radioactive atoms that decayed in a day.

b. Find the probability that on a given day, exactly 50 radioactive atoms decayed.

The total number of atoms that decayed in a year is equal to 22713.

a.

The total number of atoms that decayed in the year is given to be equal to 22713.

The total number of days in the year is equal to 365.

The mean number of atoms that decayed per day is equal to:

\(\begin{aligned}{c}\mu = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{atoms}}\;{\rm{decayed}}\;{\rm{in}}\;{\rm{the}}\;{\rm{year}}}}{{{\rm{Number}}\;{\rm{of}}\;{\rm{days}}\;{\rm{in}}\;{\rm{a}}\;{\rm{year}}}}\\ = \frac{{22713}}{{365}}\\ = 62.2\end{aligned}\)

The mean number of atoms decayed per day is equal to 62.2.

b.

Let X be the number of atoms that decayed per day.

Here, X follow a Poisson distribution with mean equal to\({\kern 1pt} \mu = 62.2\).

The probability that exactly 50 atoms get decayed in a day is computed below:

\(\begin{aligned}{c}P\left( x \right) = \frac{{{\mu ^x}{e^{ - \mu }}}}{{x!}}\\P\left( {50} \right) = \frac{{{{\left( {62.2} \right)}^{50}}{{\left( {2.71828} \right)}^{ - 62.2}}}}{{50!}}\\ = 0.0156\end{aligned}\]

Therefore, the probability that exactly 50 atoms get decayed in a day is equal to 0.0156.