Probability of winning the series given not playing the first game

Question

Suppose three pool players A, B, C are equally good, meaning that they each have a 0.5 probability of winning against each other. They are playing a game as follows: the winner of a game plays whoever didn't play the last round. For example, if A plays B, and B loses, then A plays C. Whoever wins 3 games first wins the series.

The question is, what is the probability of A winning the series, given that A does not play the first game?

It is clear that A is in disadvantage. Consider a smaller series: whoever wins 1 game first, then A has no chance of winning.

I have run some simulations in python, and found that the probability of A winning a 3-game series is about 0.25, and the other two players equally share the remaining 0.75. I'm wondering how to do this calculation mathematically.

def play():
    t = rr(0,1) # randomly picks 0 or 1
    p = {1 : 0, 2 : 0, 3 : 0} # tracks each player's number of wins
    all = {1,2,3} # all players
    match = {2,3} # players in match currently

    # player 2 and 3 play first
    p[2] += t
    p[3] += 1 - t
    winner = 2 if t == 1 else 3

    while max(p.values()) < 3:
        t = rr(0,1)
        remaining = (all - match).pop() # whoever does not play in the game
        match = {winner, remaining} # winner of the match plays the remaning player
        p[winner] += t
        p[remaining] += 1 - t
        winner = winner if t == 1 else remaining

    return max(p, key = p.get)enter code here

wins = np.array([0,0,0])
for ii in range(int(1e6)):
    wins[play() - 1] += 1
print(wins / sum(wins))
#[0.250308 0.375203 0.374489]

P.S. This question arose when I lost the same game, and I did not play the first game. When I lost a game and the score became 2-2-2, I realized that I had no chance of winning.

Answer 1

WLOG let it be that $B$ wins the first match.

Then by a victory of $A$ only the folowing sequences of winners are possible:

• $BAAA$
• $BAABCA$
• $BACBAA$
• $BACCAA$
• $BBCAAA$

So the probability on a victory for $A$ is: $$\frac18+4\times\frac1{32}=\frac14$$