Creating fair odds with a biased coin

This question asks how to obtain a genuinely fair, binary decision from a coin whose bias is unknown and possibly extreme. The setup sits in the classic family of coin-flipping constructions, where you are not allowed to adjust or recalibrate the coin, only to choose how to use its repeated flips. It is a staple of probability and algorithmic thinking interviews, and often appears in quant research and trading interviews that emphasize probabilistic intuition and clean reasoning about randomness and independence.

To answer it well, a candidate needs a solid grasp of conditional probability, independence of repeated trials, and how to exploit symmetry in sequences of outcomes. The interviewer is looking for recognition that individual flips can be biased yet certain composite events must still occur with equal probability. They also watch for the ability to design a procedure that terminates almost surely, reason about its correctness without knowing explicit parameters, and articulate why the constructed outcomes are provably fair rather than just numerically close.

What it tests

When faced with a process where the basic event is biased or unknown, fairness can often be restored by constructing composite events whose probabilities are guaranteed to be equal by symmetry, regardless of the underlying bias. The key is to identify pairs (or sets) of outcomes that are mirror images under the bias, so their probabilities must match even if the individual probabilities do not. This is possible because, for independent trials, the probability of a sequence is the product of the individual probabilities, and swapping the order of distinct outcomes (like heads then tails vs. tails then heads) yields equal probabilities. The principle holds because the structure of independence and the sum-to-one constraint force certain symmetries in the joint distribution, even when the marginals are unknown or unequal.