A fair six-sided die is rolled repeatedly

Question

I need help with this homework problem.

A fair six-sided die is rolled repeatedly. Each time the die is rolled, the number showing is written down. Let $X$ be the number of rolls until some number shows twice. The possible values of $X$ are $2,3,4,5,6,$ or $7$. For each of the following parts, explain your reasoning and express your answer as a fraction in reduced form.

Here is the solution I tried to derive.

• $\Pr(3) = \dfrac{30}{216}$

$$6 \times (6-1) \times 1 = 30$$ $$6^3 = 216$$

• $\Pr(4) = \dfrac{120}{1296}$

$$6 \times (6-1) \times (6-2) \times 1 = 120$$ $$6^4=1926$$

• $\Pr(5) = \dfrac{360}{7776}$

$$6 \times (6-1) \times (6-2) \times (6-3) \times 1 = 360$$ $$6^5=7776$$ Can someone help me out please? Thank

Answer 1

I'll let $P(n)$ denote the probability of $X = n$ for brevity. If you want the distribution of $X$, then it's clear that $P(2) = 1/6$, and there's a recurrence $$P(n+1) = \frac{n}{6}\left( 1 - \sum_{i=2}^n P(i) \right) $$ because the probability that the first $n$ rolls have no repeats is $1 - \sum_{i=2}^n P(i)$ and the probability that the $(n+1)$th roll repeats one of the first $n$ is $n/6$. You can tabulate the values of $P(n)$ by hand without much trouble.

Answer 2

You are on the right track, but the last roll of the die could be any of the numbers already rolled. Then $$\text{Pr}(2)=\frac66\times\frac16=\frac{6}{36}=\frac{1}{6}$$ $$\text{Pr}(3)=\frac66\times\frac{(6-1)}{6}\times\frac26=\frac{60}{216}=\frac{5}{18}$$ $$\text{Pr}(4)=\frac66\times\frac{(6-1)}{6}\times\frac{(6-2)}{6}\times\frac36=\frac{360}{1296}=\frac{5}{18}$$ $$\text{Pr}(5)=\frac66\times\frac{(6-1)}{6}\times\frac{(6-2)}{6}\times\frac{(6-3)}{6}\times\frac46=\frac{1440}{7776}=\frac{5}{27}$$ $$\text{Pr}(6)=\frac66\times\frac{(6-1)}{6}\times\frac{(6-2)}{6}\times\frac{(6-3)}{6}\times\frac{(6-4)}{6}\times\frac56=\frac{3600}{46656}=\frac{25}{324}$$ $$\text{Pr}(7)=\frac66\times\frac{(6-1)}{6}\times\frac{(6-2)}{6}\times\frac{(6-3)}{6}\times\frac{(6-4)}{6}\times\frac{(6-5)}{6}\times\frac66=\frac{4320}{279936}=\frac{5}{324}$$

Note that $\text{Pr}(2) + \text{Pr}(3) + \dots + \text{Pr}(7)=1$

Hope it help.