Solutions - 5110-08

(a)

$L$ follows a Pascal distribution with parameters $p=0.2$ and $\ell =5$.

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(b)

Compute $P_L(10)$:

$$\begin{array}{c}{P}_L(10)=\left(\genfrac{}{}{0px}{}{k-1}{\ell -1}\right){(1-p)}^{k-\ell }{p}^{\ell }\\ \\ \gg \gg \left(\genfrac{}{}{0px}{}{9}{4}\right){(0.8)}^5{(0.2)}^5\gg \gg \approx 0.0132\end{array}$$

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(c)

Compute $P[L\ge 7]$:

The minimum number of passersby before a winner is declared is $5$, so $P[L\ge 7]=1-P_L(5)-P_L(6)$.

$$\begin{array}{c}P[L\ge 7]=1-P_L(5)-P_L(6)\\ \\ \gg \gg 1-\left(\genfrac{}{}{0px}{}{4}{4}\right){(0.8)}^0{(0.2)}^5-\left(\genfrac{}{}{0px}{}{5}{4}\right){(0.8)}^1{(0.2)}^5\gg \gg 0.9984\end{array}$$