PLEASE I NEED THIS RN
Question 5: 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. = 5. Let A be the event that di is the lowest die out of {d, d2, d3, d4} and that d = i. Let X(W) = d2 +d3 +d4 if we As and let X}(W) = 0 if w A. Find E(X{(w)) given that WE A. That is, assuming that i is the lowest die roll and d = i, find the expected value of X}. Recall that VW E A, X}(w) = d2(W) +d3(W) +d4(w), where each of d2, d3, and d4 is at least i. 6. Observe that the term E(X:(w)) in Ewen E(X{(w)) is a constant, since it is the weighted average of the values VW E Ai, X;(w). Similarly the term E(X{(W)) in Ewea: 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)) (3(3 E(Y) > (3. (3+)) (6 i + 1)4 (6 - 1) 64 i=1 which, if you plug into Wolfram alpha, is > 11.63. Bonus: In part 6 we provide that E(Xi) > 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 CA;. Question 5: 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. = 5. Let A be the event that di is the lowest die out of {d, d2, d3, d4} and that d = i. Let X(W) = d2 +d3 +d4 if we As and let X}(W) = 0 if w A. Find E(X{(w)) given that WE A. That is, assuming that i is the lowest die roll and d = i, find the expected value of X}. Recall that VW E A, X}(w) = d2(W) +d3(W) +d4(w), where each of d2, d3, and d4 is at least i. 6. Observe that the term E(X:(w)) in Ewen E(X{(w)) is a constant, since it is the weighted average of the values VW E Ai, X;(w). Similarly the term E(X{(W)) in Ewea: 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)) (3(3 E(Y) > (3. (3+)) (6 i + 1)4 (6 - 1) 64 i=1 which, if you plug into Wolfram alpha, is > 11.63. Bonus: In part 6 we provide that E(Xi) > 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 CA
