Three standard fair 6-sided dice are rolled. What is the expected number of 6's rolled? (For example, if two 6's and a 5 are rolled, then the answer is 2.)
+195 Let X be the random variable that represents the number of 6's rolled when three fair 6-sided dice are rolled. We can define X as the sum of three indicator variables, Xi, where Xi is 1 if the ith die rolls a 6 and 0 otherwise.
Then, the expected value of X is:
E(X) = E(X1 + X2 + X3)
= E(X1) + E(X2) + E(X3)
Since each die is fair, the probability of rolling a 6 on any given roll is 1/6. Therefore, the expected value of each Xi is:
E(Xi) = 1/6 * 1 + 5/6 * 0
= 1/6
Substituting this into the equation for E(X), we get:
E(X) = E(X1) + E(X2) + E(X3)
= 1/6 + 1/6 + 1/6
= 3/6
= 1.5
Therefore, the expected number of 6's rolled when three fair 6-sided dice are rolled is 1.5.
