Determine expected winnings if a game is played where a fair coin is tossed until the first tail occurs

Question

A game is played where a fair coin is tossed until the first tail occurs. The probability $x$ tosses will be needed is $f(x) = 0.5^x$, $x = 1,2,3,\dots\;$. You win $2^x$ if $x$ tosses are needed for $x = 1, 2, 3, 4, 5$ but lose $256$ if $x > 5$. Determine your expected winnings

Answer 1

HINT: Your expected winnings are just the weighted sum of your possible winnings, the weight of a given winning being the probability of that winning. You have:

$$\begin{array}{cccc} \text{tosses}&\text{winnings}&\text{probability}&\text{weighted winnings}\\ \hline 1&2&\frac12&1\\ 2&4&\frac14&1\\ 3&8&\frac18&1\\ 4&16&\frac1{16}&1\\ 5&32&\frac1{32}&1\\ \hline 6&-256&\frac1{64}&?\\ 7&-256&\frac1{128}&?\\ \vdots&\vdots&\vdots&\vdots \end{array}$$

The weighted sum of the first five rows is $5$. Set up an infinite series for the weighted sum of the rest, evaluate it, and add to $5$ to get the expected winnings.

Answer 2

Hint: Let $X$ denote the random variable that indicates the number of tosses the game goes.

Your expected winnings are $$Pr(X=1)2^1+Pr(X=2)2^2+Pr(X=3)2^3+Pr(X=4)2^4+Pr(X=5)2^5+Pr(X>5)(-256)$$

To find $Pr(X>5)$, use the fact that $Pr(X>5)=1-Pr(X=1)-Pr(X=2)-Pr(X=2)-Pr(X=3)-Pr(X=4)-Pr(X=5)$.