SUMMARY
The Coin-Flip Puzzle: Why the Answer Is 50%
Source: Presh Talwalkar — MindYourDecisions (YouTube)  |  Type: Mathematics / Probability
Explainer  |  Context: Analysis of a banned Google interview question
The Problem
Business Insider once published a list of 15 banned Google interview questions. One was a
probability riddle: in a fictional country where families have children until they have a boy
(stopping at the first boy), what is the expected ratio of boys to girls? The question was
likely banned for its gendered phrasing and its irrelevance to job competency, but it contains
an interesting mathematical core.
Rephrased neutrally: every subject flips a fair coin until it lands heads, then stops. Tails
means flip again. What is the expected percentage of heads across all flips?
The Wrong Intuition
The first instinct is that tails should far outnumber heads, since each subject produces exactly
one heads but potentially many tails.
The Common Mistake
Listing the possible outcomes — H, TH, TTH, TTTH, TTTTH — and truncating at five
events gives 5 heads out of 1+2+3+4+5 = 15 total flips, suggesting a ratio of 5/15 = 1/3.

This is wrong. The calculation treats each outcome as equally likely, ignoring that H
(probability 0.5) is far more common than TTTTH (probability 0.5
5
). Without
probability weighting, the count is a biased estimator.
Correcting with Probability Weights
With 800 subjects, the expected distribution is: 400 end in one flip, 200 in two, 100 in three,
50 in four, and 25 in five. The total flip counts are 400, 400, 300, 200, and 125 respectively.
All 800 subjects produce one heads each, so the ratio is 800/1,425 ≈ 56.1% — already close
to 50%, with the gap closing as more terms are included.
The Intuitive Explanation
Imagine all subjects flipping simultaneously in rounds. In round 1, all n subjects flip — half
get heads and stop, half get tails and continue. Exactly 50% of round 1's flips are heads. In
round 2, the n/2 survivors flip again — once more, 50% are heads. Every subsequent round
has the same 50/50 split. Since each round is internally 50% heads, the aggregate across all
rounds is also 50%.
Key point: The shrinking population changes the volume of flips per round but not the
composition. Every round is a set of fair coin flips, so every round is 50% heads.
The Rigorous Proof
With n subjects, the total number of heads is n (one per subject). The total number of flips F
is the weighted sum: F = 1·(0.5)n + 2·(0.5)
2
n + 3·(0.5)
3
n + ...

Using the shift-and-subtract technique: compute 0.5F by shifting all terms one position, then
subtract from F. The result collapses into a geometric series summing to n, giving 0.5F = n,
and therefore F = 2n.
Heads / Total flips = n / 2n = 1/2 = 50%.