$b^{*}(10 ; 5,0.2)=\left(\begin{array}{c}10-1 \\ 5-1\end{array}\right) 0.2^{5} \times 0.8^{10-5}$

Step 1 ：The problem is asking to calculate the binomial probability. The binomial probability formula is given by: \(b^{*}(n ; k,p)=\left(\begin{array}{c}n \\ k\end{array}\right) p^{k} \times (1-p)^{n-k}\) where: \(n\) is the number of trials, \(k\) is the number of successful trials, \(p\) is the probability of success on each trial.

Step 2 ：In this case, we have \(n=10\), \(k=5\), and \(p=0.2\). We can substitute these values into the formula to calculate the binomial probability.

Step 3 ：Substituting the values into the formula, we get: \(b^{*}(10 ; 5,0.2)=\left(\begin{array}{c}10-1 \\ 5-1\end{array}\right) 0.2^{5} \times 0.8^{10-5}\)

Step 4 ：Calculating the binomial coefficient, we get: \(\left(\begin{array}{c}10-1 \\ 5-1\end{array}\right) = 126\)

Step 5 ：Calculating the binomial probability, we get: \(0.2^{5} \times 0.8^{10-5} = 0.013212057600000006\)

Step 6 ：Final Answer: The binomial probability \(b^{*}(10 ; 5,0.2)\) is approximately \(\boxed{0.0132}\)