Dice problem, with image explanation.

Question

Four dice are rolled SIMULTANEOUSLY. $X_1$ is a random variable that represents how many 1's you get in a single roll. $X_2, ... , X_4$ are defined in the same fashion, like in the picture below.

Remember the rolls are NOT consecutive. Let Y = $X_1X_2X_3X_4$. What values Y can get? Get $E(Y)$

Answer 1

$Y$ takes the values $0$ and $1$, the latter if and only if you have one each of $1,2,3$ and $4$, which occurs with probability $4!/6^4=1/54$, which is thus the value of $E(Y)$.

Answer 2

The variable $Y=0$ unless your four numbers are exactly $1,2,3,4$ in some order. In that case $Y=1$, because $X_1=X_2=X_3=X_4=1$.