A coin is considered fair if when tossed, it always comes up Heads or Tails with equal probabilities . Now consider the following gamble, based on the repeated toss of a fair coin: If the coin comes up Heads at the first toss, the gamble pays out 2 dollars, and the gamble ends. If the first toss yields Tails, the coin is tossed again, and if it comes up Heads at the second toss, the gamble pays out dollars, and the gamble ends after the second toss. If the second toss also yields Tails, a third toss is conducted, with the gamble ending with a payout of if the outcome is Heads, and the gamble moving on to a fourth toss if the outcome is Tails. The gamble continues in this fashion, by repeatedly tossing the coin until the first Heads appears, so that if the first Heads appears in the -th toss, the gamble ends with a payout of dollars. If no Heads ever appears, the gamble continues forever. Note that since the probability of getting a Heads or Tails at any toss stays equal to independently of the outcomes of previous tosses, the probability of getting a Heads in the second toss is equal to , which is the probability of getting a Tails in the first toss times the probability of getting a Heads in the second. Similarly, the probability of getting the first Heads in the -th toss is equal to . (Try to sketch a tree diagram to illustrate this gamble.) The probability that the gamble pays out less than or equal to 8 dollars (which is just the sum of the probabilities of all payouts that are smaller than or equal to 8 dollars), is equal to Answer.................. The expected payoff of the gamble is equal to A.1000 dollars B.10 dollars C.4 dollars D.infinity E.none of the other provided answers are correct A decision-maker who values the gamble based on its expected payoff, A.will not B. will C.be willing to pay 995 dollars for such a gamble. .

1. A coin is considered fair if when tossed, it always comes up Heads or Tails with equal
probabilities . Now consider the following gamble, based on the repeated toss of a fair coin: If
the coin comes up Heads at the first toss, the gamble pays out 2 dollars, and the gamble ends. If
the first toss yields Tails, the coin is tossed again, and if it comes up Heads at the second toss, the
gamble pays out dollars, and the gamble ends after the second toss. If the second toss also yields
Tails, a third toss is conducted, with the gamble ending with a payout of if the outcome is Heads,
and the gamble moving on to a fourth toss if the outcome is Tails. The gamble continues in this
fashion, by repeatedly tossing the coin until the first Heads appears, so that if the first Heads
appears in the -th toss, the gamble ends with a payout of dollars. If no Heads ever appears, the
gamble continues forever.
Note that since the probability of getting a Heads or Tails at any toss stays equal to
independently of the outcomes of previous tosses, the probability of getting a Heads in the
second toss is equal to , which is the probability of getting a Tails in the first toss times the
probability of getting a Heads in the second. Similarly, the probability of getting the first Heads
in the -th toss is equal to . (Try to sketch a tree diagram to illustrate this gamble.)
The probability that the gamble pays out less than or equal to 8 dollars (which is just the sum of
the probabilities of all payouts that are smaller than or equal to 8 dollars), is equal to
Answer..................
The expected payoff of the gamble is equal to
A.1000 dollars
B.10 dollars
C.4 dollars
D.infinity
E.none of the other provided answers are correct
A decision-maker who values the gamble based on its expected payoff,
A.will not
B. will
C.be willing to pay 995 dollars for such a gamble.