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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.)

 Mar 6, 2023
 #1
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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.

 Mar 6, 2023
 #2
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You made a small typo. The answer should be: 1/2

Guest Mar 6, 2023

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