Question

Your friend wants to make a bet with you that he or she can “roll dice better” than you can and asks that you both roll a pair of die 10 times each and compare your average outcome value to determine the winner (higher average wins). Using the concepts of probability, explain whether or not this “bet” has any actual merit.

Answer 1

Solⁿ:

No this bet does not have any actual merit. Reason being the fact that the expectation(average) of outcomes of your dice is same as average of his dice outcomes. This can be shown mathematically:

Suppose Xi (1≤i≤10) is a random variable associated to i​​​​​​ th dice roll. Xi = the value on the dice rolled. Then X = (X1 + X2 + ------ + X10)/10 would be the average outcome on the dice.

(E is expectation which is same as average)

E(Your dice) = E((X1+ X2 + ------- + X10)/10)

= (E(X1) + E(X2) + ------ + E(X10))/10 (by linearity of expectation)

= (((1/6)×1 + (1/6)×2 + ---- + (1/6)×6)×10)/10

= 3.5

Since the calculation of the expectation on your friend's dice would be entirely same, we can conclude that you and your friend have same average outcome value. So this is why there is no merit of playing this game.