## Five Same Numbers

You roll five fair, six-sided dice, all at the same time. In that one roll, what is the probability of getting five of the same number?
Source: NCTM Mathematics Teacher, September 2006

Solution
When you roll five dice at the same time, $6$ possible numbers from $1$ through $6$ can come up on each of the five dice. The number of possible outcomes = $6\times 6\times 6\times 6\times 6=6^5$. There are $6$ possible ways the five dice come up with the same number: $11111,22222,33333,44444,55555,66666$. The probability of getting five of the same number in one roll = $6/6^5=1/6^4=1/1296$.

Answer: $1/1296$

Alternative solution
Suppose the event $11111$ occurs on a roll. Since the number $1$ on the first dice is independent of the number $1$ on the second dice, etc., the probability of the event $11111=1/6\times 1/6\times 1/6\times 1/6\times 1/6=1/6^5$. The probability of getting five of the same number in one roll = $1/6^5+1/6^5+1/6^5+1/6^5+1/6^5=6(1/6^5)=1/6^4=1/1296$.