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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.