Sampling With Replacement - The Binomial Distribution
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A die is rolled three times. If \(x\) denotes the appearance of a '4' on the upper face of the die, find \({{\mathop{\rm P}\nolimits} _r}(x = 2)\)
\begin{align}
&\text{Let } \operatorname{P_r}(4)=p \text{ and } \operatorname{Pr}(\overline{4})=q\\
&\begin{aligned}
\therefore(p+q)^{3} &=p^{3}+3 p^{2} q+3 p q^{2}+q^{3} \\
3 p^{2} q &=3 \times\left(\frac{1}{6}\right)^{2} \times \frac{5}{6}=\frac{5}{72} \\
\end{aligned}\\
&\therefore \operatorname{P_r}(x=2)=\frac{5}{72}
\end{align}
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