Average of number of dice rolls untill getting specific number

Question

If we roll a dice till getting the number 5 (or any other number) - how can I calculate the average number of dice rolls?

Thanks

Answer 1

Let $a$ be the average number of rolls.

When you roll the die the first time you have probability $1/6$ of obtaining a 6 and probability $5/6$ of needing to roll again a further (on average) $a$ times.

So we have a simple equation: $$a=\frac{1}{6}.1+\frac{5}{6}.(a+1).$$

This gives $a=6$.