Suppose that 4 fair dice are rolled. Let $M$ be the minimum of 4 numbers rolled. What are the possible values of $M$. Find $E(M)$
I can't seem to get the correct answer.
The possible values of M are { $1,2,3,4,5,6$ }
$P(M=1)=\frac{1}{6}$
$P(M=2)=\frac{5^3}{6^4}$
$P(M=3)=\frac{4^3}{6^4}$
$P(M=4)=\frac{3^3}{6^4}$
$P(M=5)=\frac{2^3}{6^4}$
$P(M=6)=\frac{1}{6^4}$
Did i not calculate the probabilities correctly? I took it to be that say if $M=i$, $P(M=i)=\frac {1\cdot (6-i+1)^3}{6^4}$ by solely basing on the possible outcomes. I thought of permutating each case but that just gets me to a probability of more than 1 which is not possible. I am pretty sure my probability formula is wrong.
Best Answer
A helpful formula: If $X$ is a non-negative integer r.v., then $$ E[X] = \sum_{k\ge 1} P(X\ge k). $$ Hence $$ E[M] = \sum_{k= 1}^6 \frac{(6-k+1)^4}{6^4} = \frac{2275}{1296}. $$