Brain Teasers · The Basketball Ballot
Ballot Problem Puzzle - Did A Lead All Game? Played
Team A beat Team B 10 to 5. What's the chance A led after every single score?
What is the ballot problem?
Bertrand's ballot theorem: if A gets a votes and B gets b < a, counted in random order, A stays strictly ahead the whole way with probability (a - b)/(a + b).
The proof swaps the scores up to the first tie, which is the reflection principle used for random walks and barrier options.
How it works
A basketball game ends 10 to 5, every score worth one point, every order of the 15 scores equally likely. The story asks the probability A was strictly ahead after every score. You type your answer and check it. Then you step through the answer and the breakdown.
How scoring works
This teaser checks your answer. Any correct form counts - a fraction, a decimal rounded correctly, a percent, or the number in words. You get two tries before the answer opens. A correct first try earns a ticket; a second try or opening the answer early is recorded but earns nothing. The story teasers share one lifetime unlock for free users.
The key insight
Games where B scores first fail, and they are 5/15 = 1/3 of all orders. Failing games that start with A pair up exactly with games that start with B by swapping up to the first tie, so they are another 1/3. That leaves 1/3.
The full step-by-step solution lives at /brain-teasers/ballot-problem/solution.
A worked example
Small case: A wins 2-1. Of AAB, ABA and BAA only AAB keeps A ahead, which is 1/3, and (2 - 1)/3 = 1/3.
Common mistakes
• Answering 2/3, the chance A scores first.
• Counting ties as leading.
• Trying to count 3,003 orders by hand.
Why interviews test this
The formula is good; the swapping argument is what interviewers really want. The written solution is at /brain-teasers/ballot-problem/solution.
Play The Basketball Ballot on Outcry
The Basketball Ballot is free to play on Outcry, in the Brain Teasers section, and every run is timed and scored. Read the guide once, then play The Basketball Ballot and check your score against the strategy above. The guide book has a written guide for every other game, and the practice hub and mock assessments put the same skills under a real screen's clock.