4 Coin Flips Tail Probability

This probability question is about comparing two independent sequences of random outcomes and asking how likely they are to match in a specific statistic. Instead of caring about the exact pattern, the focus is on how many times a particular result occurs. That shift from raw outcomes to a summary count is central in quant prep, because many trading and risk problems compress complex randomness into simpler statistics.

It trains comfort with discrete distributions, independence, and combining probabilities across multiple possible matching cases. You practice turning a verbal interview setup into the right random variable, then aggregating match probabilities in a clean, systematic way. This sharpens your probabilistic thinking beyond basic "single-coin" intuition.

It matters in quant interviews because market-making, pricing, and risk all rely on understanding distributions of counts, overlaps, and coincidences between independent random processes.

What it tests

When two independent random processes each produce an outcome from the same discrete probability distribution, the probability that they produce the same outcome is the sum of the squares of the probabilities for each possible outcome. This arises because the joint probability that both select a particular outcome $k$ is the product $P(k) \times P(k)$, and the events are independent. Summing over all possible outcomes accounts for all ways they can match. This pattern holds for any pair of independent draws from the same distribution, not just coins or tails: it is a direct consequence of the definition of independence and the law of total probability. The intuition is that matching is likelier when the distribution is more concentrated (less uniform), and less likely when outcomes are more evenly spread.