In a tournament, a team plays 10 matches with probabilities of winning and losing each match as…

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\frac{1}{3}$ and $\frac{2}{3}$ respectively. Let $x$ be the number of matches that the team wins, and $y$ be the number of matches that team loses. If the probability $\mathrm{P}(|x-y| \leq$ 2) is $p$, then $3^9 p$ equals ______

Solution

$\begin{aligned} & P(W)=\frac{1}{3} \quad P(L)=\frac{2}{3} \\ & x=\text { number of matches that team wins } \\ & y=\text { number of matches that team loses } \\ & |x-y| \leq 2 \text { and } x+y=10 \\ & |x-y|=0,1,2 \quad x, y \in N\end{aligned}$ Case-I $\begin{aligned} & -I:|x-y|=0 \Rightarrow x=y \\ & \because x+y=10 \Rightarrow x=5=y \\ & P(|x-y|=0)={ }^{10} C_5\left(\frac{1}{3}\right)^5\left(\frac{2}{3}\right)^5 \end{aligned}$ Case-II : $|\mathrm{x}-\mathrm{y}|=1 \Rightarrow \mathrm{x}-\mathrm{y}= \pm 1$ $\begin{array}{|c|c|} \hline \mathrm{x}=\mathrm{y}+1 & \mathrm{x}=\mathrm{y}-1 \\ \hline\because \mathrm{x}+\mathrm{y}=10 & \because \mathrm{x}+\mathrm{y}=10 \\ \hline 2 \mathrm{y}=9 & 2 \mathrm{y}=11 \\ \hline \text{Not possible} & \text{Not possible} \\ \hline \end{array}$ $\begin{aligned} & \text { Case-III : }|\mathrm{x}-\mathrm{y}|=2 \Rightarrow \mathrm{x}-\mathrm{y}= \pm 2 \\ & x-y=2 \quad \text { OR } \quad x-y=-2 \\ & \because \mathrm{x}+\mathrm{y}=10 \quad \because \mathrm{x}+\mathrm{y}=10 \\ & x=6, y=4 \quad x=4, y=6 \\ & \mathrm{P}(|\mathrm{x}-\mathrm{y}|=2)={ }^{10} \mathrm{C}_6\left(\frac{1}{3}\right)^6\left(\frac{2}{3}\right)^4+{ }^{10} \mathrm{C}_4\left(\frac{1}{3}\right)^4\left(\frac{2}{3}\right)^6 \\ & \mathrm{p}={ }^{10} \mathrm{C}_5 \frac{2^5}{3^{10}}+{ }^{10} \mathrm{C}_6 \frac{2^4}{3^{10}}+{ }^{10} \mathrm{C}_4 \frac{2^6}{3^{10}} \\ & 3^9 \mathrm{p}=\frac{1}{3}\left({ }^{10} \mathrm{C}_5 2^5+{ }^{10} \mathrm{C}_6 2^4+{ }^{10} \mathrm{C}_4 2^6\right) \\ & =8288 \\ & \end{aligned}$

Asked in: JEE Main 2024 (04 Apr Shift 2)

Practice more Probability questions on Aicharya