Capped Probability and Expected Payoff

Question

You keep flipping a coin until you get a head. You are paid 2^(# of flips) dollars. Suppose that if you make more than 210 dollars, you will only receive 210 dollars. What is the expected payoff of this game?

I am very confused how to approach this question. Is the expected payoff only half of 210=105 because the probability of getting a head is 1/2?

Am I on the right track?

Answer 1

The suggestion of $105$ is not right. Let us look at the money you make.

With probability $1/2$, you get a head immediately, payoff $2$.

With probability $1/4$, you get tail then head, payoff $4$.

With probability $1/8$, you get two tails then head, payoff $8$.

Continue $\dots$.

With probability $1/128$, you get 6 tails then head, payoff $128$.

With probability $1/128$, you get $7$ or more tails, payoff $210$.

Now you can compute the expectation.

Answer 2

Play the game $256$ times. The payoff table is:

$$\begin{array} {c|c} result&\#&payoff\\ \hline H&128&256\\ TH&64&256\\ T^2H&32&256\\ T^3H&16&256\\ T^4H&8&256\\ T^5H&4&256\\ T^6H&2&256\\ T^{7+}H&2&420 \end{array} $$

Expected payoff per game is (total payoff)/256 $\approx 8.64$