Three-Way Playoff Win Probability

Compute each player's chance of winning a three-way playoff where two players start and the third waits, and the title goes to the first player to win two matches in a row.

Input

Compute each player''s chance of winning a three-way playoff, where two players fight first, one waits, and the first to win two matches in a row takes the title. Enter the win rate for each match.

Win rate per match

Probability that the previous match winner wins the next match. 50 means evenly matched.

Result

Each player''s chance of winning

Starter A

35.7%

Starter B

35.7%

Waiter C

28.6%

Each starter

35.7 percent

Waiter

28.6 percent

Theory at 50 percent

5 to 5 to 4 (out of 14)

How it works

A and B fight the first match while C waits. The first player to win two matches in a row wins.

The winner plays the waiter next, the loser steps aside to wait, and this repeats until someone wins twice in a row.

The waiter lacks the head start of an early win, so even with equal skill the waiter is slightly behind.

How it works

The playoff has three players A, B, and C. A and B fight the first match while C waits. The first player to win two matches in a row takes the title.
If p is the chance the previous winner keeps winning and q is 1 minus p, the first-match winner takes the title with probability p divided by (1 minus q cubed).
At a 50 percent match win rate, the two starters each win with probability 5/14 (about 35.7 percent) and the waiter with 4/14 (about 28.6 percent), so the waiter is slightly behind.
The waiter is at a disadvantage because they must beat someone twice in a row to win and lack the head start of an early victory.
The three win probabilities always add up to 1 (100 percent).

Reviews

Tell us what you think of this calculator.