Suppose you visit your local casino after classes one day, and find there is a n

Question

Suppose you visit your local casino after classes one day, and find there is a new game being offered. The game works as follows. The dealer tosses a fair coin. If it comes up heads, you are paid 2 dollars and the game is over. If it comes up tails, the dealer tosses the coin again. If it comes up heads this time, you are paid 4 dollars and the game is over. If it comes up tails, the dealer tosses the coin again. If it comes up heads this time you are paid 8 dollars and the game is over. If it comes up tails, the dealer tosses the coin again, continuing in this fashion until the coin comes up heads. If the coin comes up heads on the kth toss, you win 2k dollars, otherwise, the game continues one more round.

(a) What is the probability that the game ends at the kth toss? (3 marks)

(b) What is the expected amount of money a gambler would win by playing this game? (3 marks)

(c) How much would you pay to play this game? Why? (2 marks)

Explanation / Answer

a) The game will stop whenever the head comes up and the amount paid is twice the number of tosses made. However, the coin is tossed again if a tail coes up and the coin is tossed till a head comes. The probability that the ends at the kth toss is the probability that a head comes up with the kth toss. Since upon tossing a coin the proability of gettinga head is 0.5, the probability of the game ending at the kth toss is also 0.5.

b) If the game stops at the first toss, the expected value is the probability of head and the amount won if a head comes up

Therefore, the expected value = 0.5*2 = 1

Therefore, if the game continues for k tosses, then the expected value till the (k-1)th toss is zero, since the game continues only if he gets a tail. It is only in the kth toss that he get 2k with a probability of 0.5. Therefore the expected value of money won is 0*0.5 + .................+ 0.5*2k = k.

Therefore, the expected amount won through k tosses is k.

c) A risk averse person will pay upto 2 dollars to play this game since that is the certain amount he will win by playing the game even if the game stops after the first toss if a head comes up. If a tail comes up in the first toss, he will only win more than 2 dollars in the subsequent tosses if a head comes later. Since the certain amount gained from the game is 2, he will be willing to pay 2 dollars for playing the game.

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