Question

A bag contains 7 green, 4 white and 5 red balls. If four balls are drawn one by one with replacement, what is the probability that one is red?

Answer 1

Let X denote the number of red balls drawn from 16 balls with replacement.
X follows a binomial distribution with n = 4,$p=\frac{5}{16},q=1-p=\frac{11}{16}$

$$\begin{gathered} P(X=r)={}^{4}C_r{\left(\frac{5}{16}\right)}^r{\left(\frac{11}{16}\right)}^{4-r} \\ P(\text{O}\mathrm{ne}\mathrm{ball}\mathrm{is}\mathrm{red})=P(X=1) \\ ={}^{4}C_1{\left(\frac{5}{16}\right)}^1{\left(\frac{11}{16}\right)}^{4-1} \\ =4\left(\frac{5}{16}\right){\left(\frac{11}{16}\right)}^3 \\ =\frac{5}{4}{\left(\frac{11}{16}\right)}^3 \end{gathered}$$