[Math] A six-sided die is rolled five times. What is the probability that only the final roll will be a deuce

Question

A six-sided die is rolled five times. What is the probability that only the final roll will be a deuce?

I've tried to reason this out myself but I can only think that there's a 1/6 chance that the roll will be 2 and another 1/5 chance for it to be the last one. What am I missing here?

Thanks!

Answer 1

Allow me to quote your reasoning:

1/5 chance for it to be the last one

Here is what is wrong with your reasoning: it is not guaranteed that only one roll will be a deuce. It is also possible that there are two deuces rolled.

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We need the first four rolls to be not deuce and the last one to be a deuce. Each roll is independent, so we can multiply the probabilities for each roll together.

The probability for a roll to be a deuce is $\dfrac16$, and similarly the probability for a roll to be not a deuce is $\dfrac56$. Therefore, the required probability is $\left(\dfrac56\right)^4\left(\dfrac16\right) = \dfrac{625}{7776}$.