Biased Coin Interview Questions
How to solve biased coin probability and expected value questions in quant interviews, including likelihoods, Bayes updates, and expected waiting times.
Candidates extending fair-coin methods to uneven probabilities.
Replace symmetry with parameters
Biased coin questions remove the comfort of 50-50 symmetry. Let p be the probability of heads and 1 - p the probability of tails. Carry the parameter until the end unless the prompt gives a number.
Fixed flips
For fixed flips, use binomial reasoning: choose the positions of heads and multiply by p for each head and 1 - p for each tail. Check that the fair-coin case reduces to the familiar answer.
Expected waiting
For first heads, the expected number of flips is 1/p when each flip is independent with heads probability p. The fair-coin result of 2 is the special case p = 0.5.
Concrete example
If heads has probability 0.3, the probability of exactly two heads in three flips is C(3,2)(0.3)^2(0.7). The positions matter for counting, but each chosen sequence has the same probability.
Bayes variants
Some biased coin prompts hide which coin was chosen. Then you need priors and likelihoods. Convert to a tree or table so you do not confuse P(flips | coin) with P(coin | flips).
Common mistakes
Candidates often use fair-coin symmetry after the prompt says biased, or forget that tails probability is 1 - p. Write the probabilities explicitly before counting.
Practice the pattern
Use the LeetQuidity curriculum and calibration to turn this topic into a focused practice plan.