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.)

Guest Mar 6, 2023

#1**0 **

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.

Justingavriel1233 Mar 6, 2023