Expected Value Homework Solutions 1. You roll two dice. What is ...

Answer: We have the following table for probabilities of when the first heads appears.P (0) = 1/32 The probability that no heads appearP (1) = 1/2P (2) = 1/4P (3) = 1/8P (4) = 1/16P (5) = 1/32.Thus the expected value of what he pays me back is0×P (0)+1×P (1)+2×P (2)+4×P (3)+8×P (4)+16×P (5) = 0+1/2+1/2+1/2+1/2+1/2 = 2.5.Thus my expected value is $(2.5-2), or 50 cents.11. In the game of birdcage, you pay $1 to play the game. If one of the three dice comes up 6, you get$2 back (earning $1). If two dice come up 6, you get $3 back, and if all three dice come up 6, youget $4 back. What is the expected value of the game?Answer: The probability of no sixes coming up is 125216, and in this case you lose $1. We next needthe probability of exactly one 6 coming up. This can happen in one of three ways: the six can beon the first die, the second die, or the third die. For exactly one six, with it on the first die, thereare 1 × 5 × 5 = 25 ways for this to happen. Similarly, there are 25 ways for the six to appear onthe second die, and 25 ways for it to appear on the third die (and for exactly one 6). Thus thereare 75 ways for exactly one six to appear. The value of this outcome is $1. For exactly two sixes toappear, there are 5 ways for the first two dice to be sixes, 5 ways for the first and the third dice tobe sixes, and 5 ways for the last two dice to be sixes. Consequently there are 15 ways for two diceto come up six. The value of this outcome is $2. Finally, there is exactly one way for three sixes tocome up, with a value of $3. Thus the expected value (for you) is125 75 15(−1) + (1) +216 216 216 (2) + 1 −17(3) =216 216 .Hence, for every game you play, you expect to lose 7.8 cents.12. The game “Who Wants to be a Millionaire” has $100, $200, $500, $1000, $2000, $4000, $8000,$16000, $32000, $64000, $125000, $250000, $500000, and $1000000 questions. You start at the $100levels and work your way up through the questions if you continue to get right answers. Supposeyou got to keep the amount of money of your last correct question (which isn’t what happens inthe show). Each question is multiple choice with 4 possible answers, only one of which is correct.What is the expected amount of money that a contestant would get if they guess randomly on eachquestion?Answer: We have the following probabilities for money amounts