Question

A bank plays the following game against a casino, for which the bank pays the casino a price $x to start playing. A coin is tossed repeatedly until it lands heads up. If heads appear on the first toss, the casino pays the bank $2, if head appears on the second toss, the casino pays the bank $4, if head appears on the third toss, the casino pays the bank $8 and so on.

How much should the bank be willing to pay the casino to start playing the game?

PROPOSITION A: The bank should play no matter how high the price $x is.

PROPOSITION B: The bank should only play if the price to start, $x, is less than the finite amount $.........

Answer 1

P(Heads appears on the nth toss) =

2 7l

The casino pays $ $2^{n}$ to the bank if heads appears on the nth toss

Thus, expected value of the bank's winnings

= $\sum 2^{n}*\frac{1}{2^{n}}$ for n = 1,2,3,....... inf.

The sum tends to infinity

Thus, the bank should play no matter how high the price $x is.

Answer 2

bad question not good

Answer 3

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