A single die is rolled 7 times. What is the probability that a six is rolled exactly once, if it is known that at least one six is rolled?

Question

A single die is rolled 7 times. What is the probability that a six is rolled exactly once, if it is known that at least one six is rolled?

Could you walk me through the concepts?

Answer 1

To calculate $P(\text{exactly one } 6) $, the denominator should be the total number of outcomes, which is $6^7$. Looking at the outcomes with one $6$, there are seven slots to put the 6. From the remaining six slots, there are $5^6$ possible outcomes. Hence, $$ P(\text{exactly one } 6)=\frac{7\times5^6}{6^7}. $$ For $P(\text{at least one } 6)$, use the complement law. The answer is then \begin{align} P(\text{exactly one }6|\text{at least one }6)&=\frac{P(\text{exactly one }6\text{ and at least one }6)}{P(\text{at least one }6)} \\ &=\frac{P(\text{exactly one }6)}{P(\text{at least one }6)}. \end{align}