Question

Five dice are thrown simultaneously. If the occurrence of 3, 4 or 5 in a single die is considered a success, find the probability of at least 3 successes.

Answer 1

Let X denote the occurrence of 3,4 or 5 in a single die. Then, X follows binomial distribution with
n=5.
Let p=probability of getting 3,4, or 5 in a single die .

p = $\frac{3}{6}=\frac{1}{2}$

$$\begin{gathered} q=1-\frac{1}{2}=\frac{1}{2} \\ P(X=r)={}^{5}C_r{\left(\frac{1}{2}\right)}^{\mathrm{r}}{\left(\frac{1}{2}\right)}^{5-r} \\ \mathrm{P}(\mathrm{at}\mathrm{least}3\mathrm{successes})=P(X\ge 3) \\ =P(X=3)+P(X=4)+P(X=5) \\ ={}^{5}C_3{\left(\frac{1}{2}\right)}^3{\left(\frac{1}{2}\right)}^{5-3}+{}^{5}C_4{\left(\frac{1}{2}\right)}^4{\left(\frac{1}{2}\right)}^{5-4}+{}^{5}C_5{\left(\frac{1}{2}\right)}^5{\left(\frac{1}{2}\right)}^{5-5} \\ =\frac{{}^{5}C_{\mathit{3}}+{}^{5}C_4+{}^{5}C_5}{2^5} \\ =\frac{1}{2} \end{gathered}$$