# Thread: One for Plato !

1. ## One for Plato !

A, B, and C play a video game in tournament mode.

In this mode, two players play a match, and the winner of the
match plays a new match against the player who was sitting out.

This continues until a player wins two matches in a row.

A and B play the first match.

If they are all equally skilled at the game,
what is the probability that C will win the tournament?

2. ## Re: One for Plato !

The sequence of winners will be either ACBACB...ACC, or BCABCA...BCC. Summed over all possible such sequences, we get the sum of two geometric sequences.

From this we derive the probability of

3. ## Re: One for Plato !

Originally Posted by Archie
From this we derive the probability of
...of WHAT ?

4. ## Re: One for Plato !

The first game, it does not matter who wins. The second game, C must win (or the tournament ends). There is a 50% chance that C will win. Then, there is a 50% chance that C will win the 3rd round (ending the tournament with C a winner).
If C loses the 3rd round (50% chance), then there is a 50% chance that whomever beat C in round 3 loses in round 4, then a 50% chance that C wins in round 5, followed by a 50% chance that C wins in round 6 (ending the tournament with C a winner).
If C loses in round 6 (50% chance), then there is a 50% chance that whomever beat C in round 6 loses in round 7, a 50% chance that C wins in round 8, then a 50% chance that C wins in round 9.

This produces the sum:

$$\dfrac{1}{2^2}+\dfrac{1}{2^5}+\dfrac{1}{2^8}+ \cdots = \dfrac{1}{4}\sum_{n\ge 0}{\left(\dfrac{1}{8}\right)^n} = \dfrac{2}{7}$$

A and B each have a $\dfrac{5}{14}$ chance of winning the tournament.

5. ## Re: One for Plato !

Originally Posted by DenisB
...of WHAT ?
... me completing the sentence?