Question

There are $6$ boxes labeled $B_1,B_2,....B_6$. In each trial, two fair dice $D_1,D_2$ are known. If $D_1$ shows $j$ and $D_2$ shows $k$, then $j$ balls are put into the box $B_k$. After $n$ trials, what is the probability that $B_1$ contains at most one ball?

• A. $\left(\frac{5^{n-1}}{6^{n-1}}\right)+\left(\frac{5^n}{6^n}\right)\left(\frac{1}{6}\right)$
• B. $\left(\frac{5^n}{6^n}\right)+\left(\frac{5^{n-1}}{6^{n-1}}\right)\left(\frac{1}{6}\right)$
• C. $\left(\frac{5^n}{6^n}\right)+n\left(\frac{5^{n-1}}{6^{n-1}}\right)\left(\frac{1}{6}\right)$
• D. $\left(\frac{5^n}{6^n}\right)+n\left(\frac{5^{n-1}}{6^{n-1}}\right)\left(\frac{1}{6^2}\right)$

Answer 1

Answer: (D)

$\left(\frac{5^n}{6^n}\right)+n\left(\frac{5^{n-1}}{6^{n-1}}\right)\left(\frac{1}{6^2}\right)$

Let $P\left(B_1\right)$ be the probability that $B_1$ contains atmost one ball

$P\left(B_1\right)=P(B_1$ contain $0$ balls) $+P(B_1$ contains $1balls)$

$=P(D_1$ never show $1)+P(D_2$ shows $1$ once when $D_1$ show $1)$

$={\left(\frac{5}{6}\right)}^n+\{{}^n{C}_1.{\left(\frac{5}{6}\right)}^{n-1}.\frac{1}{6}\}.\frac{1}{6}\Rightarrow$ Option $D$.