A pair of dice is thrown 5 times.

Question

A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is $$\frac{k}{3^{11}}$$, then $k$ is equal to :

Accepted Answer

Given, a pair of dice is thrown once, then number of outcomes $=6 \times 6=36$

A total of 5 are $\{(1,4),(2,3),(4,1),(3,2)\}$

$$
\begin{aligned}
& \therefore p  =p(\text { success })=\frac{4}{36}=\frac{1}{9} \\
& q  =1-\frac{1}{9}=\frac{8}{9}
\end{aligned}
$$

Let $X$ represent the number of success

$$
\begin{aligned}
& \therefore P(\text { at least } 4 \text { success }) \\
&=P(X=4)+P(X=5) \\
&={ }^5 C_4 p^4 q+{ }^5 C_5 p^5 q^{5-5} \\
&=5\left(\frac{1}{9}\right)^4\left(\frac{8}{9}\right)+\left(\frac{1}{9}\right)^5 \\
&=\frac{40+1}{\left(3^2\right)^5}=\frac{41}{310} \\
&=\frac{41 \times 3}{3^{11}}=\frac{123}{3^{11}} \\
& \therefore k=123
\end{aligned}
$$

Solutions · 1