Bertrand's Ballot Problem

The problem
A wins 7 votes · B wins 3 votesA7 votesB3 votesvotes counted in random orderP(A always strictly ahead) = ?

In an election, candidate A receives aa votes and B receives bb votes, with a>ba > b. The a+ba + b votes are counted in a uniformly random order.

What is the probability that A is strictly ahead of B throughout the entire count?

(Concretely: for a=7a = 7, b=3b = 3, what's the answer?)

Tempting (but wrong)
"depends on the ordering — too many to enumerate?"A leadsB leadstied

The problem looks combinatorially heavy:

  • (103)=120\binom{10}{3} = 120 possible orderings of 7 A's and 3 B's.
  • For each, you'd need to check if A is ever tied or behind.
  • Brute force is doable but offers no insight.

People also sometimes guess "A always leads because A wins big," underestimating how often early B-votes can produce a tie at 11-11 or 22-22.