Calculators
Ballot Probability Calculator
Calculate the exact chance that candidate A stays strictly ahead throughout a random two-candidate vote count.
Enter 0 to 1000 votes per candidate, with at least one vote overall. Count sequences where A leads after every vote, including the first and last. Ties at any point fail. Each ordering of A/B votes is equally likely. Results stay in your browser.
How strict-lead ballot probabilities work
For a votes for A and b votes for B, all C(a + b, a) A/B sequences are equally likely. If a > b, the ballot theorem gives a strict-lead probability of (a − b)/(a + b). Multiplying this fraction by the total sequence count gives the exact favorable count. If a ≤ b, a strict lead through the final vote is impossible, so the probability is zero. A tie at any stage is excluded; this is different from allowing A to stay tied or ahead. For A = 5 and B = 3, there are 56 sequences and 14 favorable sequences, giving 1/4 or 25%. If only A receives votes, the probability is 1; if only B receives votes, it is 0. The empty vote count is excluded. Enter whole numbers from 0 to 1000 for each candidate. Signs, decimals, separators and scientific notation are rejected; leading zeros and surrounding whitespace are accepted. BigInt arithmetic preserves exact sequence counts and the reduced probability fraction. The displayed percentage is rounded to 12 significant digits. This is a combinatorial model with fixed totals and uniformly random orderings, not an election forecast. All calculations stay in your browser.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
Does a tie count as a strict lead?
No. A must have more votes than B after every counted vote. A tie at any point makes the sequence unfavorable.
What happens if A has no final majority?
The probability is zero when A has at most as many votes as B, because A must also lead after the final vote.
What assumptions does the calculator make?
The two vote totals are fixed and every ordering of A/B votes is equally likely. It does not forecast election outcomes.
Are the results exact?
Sequence counts and the reduced probability fraction use exact BigInt arithmetic. The percentage is rounded to 12 significant digits. Inputs remain in your browser.