Recitation 22 Problem 2. Here's yet another fun 6.042 game! You pick a number between l and 6. Then you roll three fair, independent dice If your number never comes up, then you lose a dollar If your number comes up once, then you win a dollar. If your number comes up twice, then you win two dollars If your number comes up three times, you win four dollars! What is your expected payoff? Is playing this game likely to be profitable for you or not? Solution. Let the random variable r be the amount of money won or lost by the player in a round. We can compute the expected value of R as follows Ex(r)=-1 Pr(0 matches)+1. Pr(1 match)+2. Pr(2 matches)+ 4. Pr(3 matches +1·3 +2.3 6/)(6/+4./1)3 -125+75+30+4 216 216 game!pect to lose 16/216 of a dollar (about 7. 4 cents)in every round. This is a horrible n eRecitation 22 6 Problem 2. Here’s yet another fun 6.042 game! You pick a number between 1 and 6. Then you roll three fair, independent dice. • If your number never comes up, then you lose a dollar. • If your number comes up once, then you win a dollar. • If your number comes up twice, then you win two dollars. • If your number comes up three times, you win four dollars! What is your expected payoff? Is playing this game likely to be profitable for you or not? Solution. Let the random variable R be the amount of money won or lost by the player in a round. We can compute the expected value of R as follows: Ex (R) = −1 · Pr (0 matches) + 1 · Pr (1 match) + 2 · Pr (2 matches) + 4 · Pr (3 matches) � �3 � � � �2 � �2 � � � �3 5 1 5 1 5 1 = −1 · + 1 · 3 + 2 · 3 + 4 · 6 6 6 6 6 6 −125 + 75 + 30 + 4 = 216 −16 = 216 You can expect to lose 16/216 of a dollar (about 7.4 cents) in every round. This is a horrible game!