Biased Coin Betting Expected Value
Biased coin betting expected value guide for quant interview prompts where head and tail probabilities are not 50/50.
Candidates who need to adapt coin-game expected value formulas when probabilities are uneven.
Change the probability model first
A biased coin prompt is not solved by reusing 50/50 intuition. Write P(heads) = p and P(tails) = 1 - p before touching payoffs.
Use net payoffs
Expected value should use the amount gained after cost if the bet wins and the amount lost if it fails. Gross prize language often hides the actual net payoff.
Concrete example
If heads has probability 0.6, heads wins 2, and tails loses 1, the expected value is 0.6 x 2 + 0.4 x -1 = 0.8.
Solve the fair price
When the prompt asks for a fair entry price, compute the expected gross payoff first. The fair risk-neutral price is the amount that makes net expected value zero.
Discuss probability sensitivity
If the edge depends on a small probability advantage, say so. A tiny error in estimating the coin bias can flip a positive-EV answer into a negative-EV one.
Common mistakes
The common failure is treating the coin label as proof of fairness. In interviews, the stated or inferred probability model controls the expected value.
Practice the pattern
Use the LeetQuidity curriculum and calibration to turn this topic into a focused practice plan.