Expected Value of a Coin Flip Game
A game pays $\$32$ if you flip 5 heads in a row using a fair coin. How much would you pay to play this game?
Hints
- What is the probability of flipping 5 heads in a row with a fair coin?
- The fair price of a game equals its expected payoff. Compute $P(\text{win}) \times \text{payout}$.
- Notice that $32 = 2^5$ and $P(5H) = (1/2)^5 = 1/32$, so the expected value is $32/32 = 1$.
Worked Solution
How to Think About It: This is a pure expected value calculation. You have a binary outcome -- either you flip 5 heads in a row and collect $\$32$, or you don't and get nothing. The fair price is just the probability of winning times the payoff. Before doing any math, notice that $32 = 2^5$, which is a strong hint that the answer will be clean.
Quick Estimate: The probability of 5 heads in a row is $(1/2)^5 = 1/32$. So the expected payoff is $32 \times 1/32 = 1$. Done -- the fair price is $\$1$.
Approach: Straightforward expected value calculation.
Formal Solution:
Each flip is independent with $P(H) = 1/2$. The probability of 5 consecutive heads is:
$$P(5H) = \left(\frac{1}{2}\right)^5 = \frac{1}{32}$$
The expected payoff is:
$$E[\text{payoff}] = \$32 \times \frac{1}{32} + \$0 \times \frac{31}{32} = \$1$$
Answer: A risk-neutral player should pay at most $\$1$ to play this game. A risk-averse player would pay strictly less than $\$1$. The fact that $32 = 2^5$ makes the payoff exactly offset the odds, giving a round-number answer.
Intuition
This problem is testing whether you can quickly compute expected value for a simple game. The key insight is that the payoff was deliberately chosen to be $2^5 = 32$, which exactly cancels the $1/32$ probability of winning. Whenever you see a payoff that looks like a power of 2, check whether it matches the number of coin flips -- the interviewer is often setting up a clean EV = $1 answer.
In practice, this kind of reasoning is the foundation of pricing binary options and digital contracts. The fair price is always $P(\text{event}) \times \text{payoff}$, and the first thing a trader does is sanity-check whether the payoff structure lines up with the odds. Risk aversion would push the price below $\$1$, but in an interview context, the risk-neutral answer is what they want.