Step 1

The binomial probability distribution is,

\(P(X=k)=\left(\begin{array}{c}n\\ x\end{array}\right)(p)^{k}(1-p)^{n-k}\)

In the formula, n denotes the number of trails, p denotes probability of success, and k denotes the number of success.

Step 2

The probability of \(\displaystyle{k}={0}\) is,

\(P(X=0)=\left(\begin{array}{c}5\\ 0\end{array}\right)(0.16)^{0}(1-0.16)^{5-0}\)

\(\displaystyle={0.4182}\)

The probability of \(\displaystyle{k}={1}\) is,

\(P(X=1)=\left(\begin{array}{c}5\\ 1\end{array}\right)(0.16)^{1}(1-0.16)^{5-1}\)

\(\displaystyle={0.3983}\)

The probability of \(\displaystyle{k}={2}\) is,

\(P(X=2)=\left(\begin{array}{c}5\\ 2\end{array}\right)(0.16)^{2}(1-0.16)^{5-2}\)

\(\displaystyle={0.1517}\)

The probability of \(\displaystyle{k}={3}\) is,

\(P(X=3)=\left(\begin{array}{c}5\\ 3\end{array}\right)(0.16)^{3}(1-0.16)^{5-3}\)

\(\displaystyle={0.0289}\)

The probability of \(\displaystyle{k}={4}\) is,

\(P(X=4)=\left(\begin{array}{c}5\\ 4\end{array}\right)(0.16)^{4}(1-0.16)^{5-4}\)

\(\displaystyle={0.0028}\)

The probability of \(\displaystyle{k}={5}\) is,

\(P(X=5)=\left(\begin{array}{c}5\\ 5\end{array}\right)(0.16)^{5}(1-0.16)^{5-5}\)

\(\displaystyle={0.0001}\)

Thus, the probability distribution given the probability \(\displaystyle{p}={0.16}\) of success on a single trial is,

\(\begin{array}{|c|c|} \hline k & P(X=k) \\ \hline 0 & 0.4182\\ \hline 1&0.3983\\ \hline2&0.1517\\ \hline3&0.0289\\ \hline4&0.0028\\ \hline5&0.0001 \\ \hline \end{array}\)