How to Price a Simple Coin Flip Game Under Interview Conditions

Pricing a coin flip game means building the probability tree branch by branch and comparing the total to the price the interviewer names.

Pricing a coin flip game under interview conditions means building the probability tree for every outcome, computing the expected payoff at each branch, and comparing the total to whatever price the interviewer names, all while talking through the steps rather than solving in silence. The skill being tested is less the arithmetic itself and more whether you can hold a tree with more than two branches in your head without losing track of a probability.

The first move is writing down every distinct outcome and its probability, not the ones that happen to pay out. A three flip game has eight equally likely sequences, but the outcomes usually collapse by the number of heads rather than the exact sequence, so it is faster to work in terms of zero, one, two, or three heads and their binomial probabilities of one eighth, three eighths, three eighths, and one eighth.

Skipping this step is the single most common reason candidates arrive at the wrong number. This style of question appears often in first round interviews at firms including Optiver and IMC Trading, where the flip count typically grows from one to four across the conversation as you answer correctly.

Consider a game with three fair coin flips. You are paid $5 for every head you flip, except that flipping zero heads costs you $2 instead of paying nothing. The four branches are: zero heads with probability one eighth paying negative $2, one head with probability three eighths paying $5, two heads with probability three eighths paying $10, and three heads with probability one eighth paying $15.

Multiplying each payoff by its probability gives negative 0.25, plus 1.875, plus 3.75, plus 1.875, which sums to $7.25. If the interviewer then asks whether you would pay $6 to play once, the answer is yes, because the expected profit is $1.25 per play, and you should say so directly rather than hedging.