In the Dungeons and Dragons question from last assignment we talked about how to roll for stats (that is, we take the sum of 3 six-sided dice). It was briefly mentioned that to get higher stats, we can roll 4 six-sided dice and take the sum of the 3 highest dice. In this question we will compare the expected outcome from both of these techniques.

_ In . Dungeons and Dragons we talked about how to roll for stats (that is, we take the sum of 3 six-sided dice). It was briey mentioned that to get higher stats, we can roll 4 six-sided dice and take the sum of the 3 highest dice. In this question we will compare the expected outcome from both of these techniques. 1. Determine the expected value of the sum of rolling 3 fair six-sided dice. 2. What is the size of the sample space S of all possible dice rolls with 4 dice? Let d1, d2, d3, d4 be random variables corresponding to the values of 4 sixsided dice after being rolled. Let Y be the value of the three highest dice. That is, Y = (2:21 di) min{d1, d2, d3, d4}. Thus our goal is to nd E(Y) and compare it to the answer froml Let the random variable Xi be the sum of the highest 3 dice if the lowest die is equal to 2', that is min{d1, d2, d3, d4} = 2'. Let X; = 0 if the lowest die is not equal to 11. Let A; be the event that the lowest die out of d1, d2, (3, and (L; is equal to 2'. 3. In a few sentences explain why 6 6 2m) = Z Esau) = Z Z Pia-(w) wES i=1 wES i=1 wEAg 4. Use the above expression and linearity of expectation to express 1930') in terms of E(X,;(w)) given that w E 14,. 5. Let A; be the event that all is the lowest die out of {d1,d2, d3, d4} and that d1 = 2'. Let X;'(w) = 612 + d3 + (4 ifw E A; and let Xg'(w) = 0 if m 95 Ag. Find E(X,f(w)) given that w E Ag. That is, assuming that i is the lowest die roll and d1 = 11, nd the expected value of X;. Recall that Vw 6 Ag, Xg'(w) = d2(w) + d3(w) + d4(w), where each of d2, 13, and d; is at least 11. 6. Observe that the term E(X,(w)) in ZweAiE(X,(w)) is a constant, since it is the weighted average of the values Vw E A,,X,;(w). Similarly the term E(X,'(w)) in Email E(X:(w)) is also a constant. We will not prove it at this time, but we will use the fact that for the expressions given above, E(X:(w)) i(3- (3%)).W i=1 which, if you plug into Wolfram alpha, is > 11.63. Bonus: In part 6 we provide that E (X,) > E (X: ) That is, the average value of the highest 3 dice of all the rolls in A, is higher than the average value of the highest 3 dice of all the rolls in A;. Explain the idea behind why this is the case. You do not need to prove it, so you may use examples to help articulate it. Hint: A; (_2 A4