probability

[Noona]

Hi,
i am squeezing my brain trying to know the answer for this question:

there are 3 players, A, B and C, they are playing a coin toss game, the game starts with A and B playing, the winner between these 2 plays with C, if he wins again, he's the winner and the game is over, if not, then the other player plays with c, the game ends only when one has won twice in a row, whats the probability A will win, B will win, c will win.
the number of games can go to infinity, in which case the prob. for anyone to win is 0,

thanks

[zaza]

1/3 each.


Draw a probability tree, and it will become obvious.


zaza

[Noona]

no it can't be 0.33 in the general case, what if the number of games is infinite? then it's 0, i believe it's got to be an X function when X is the number of the games,
i'd have thought it's 0.5*0.5*(0.5)^(k-2) k=2 3 4
but for example a didn't necessarily play in the rest of k-2 games, know what i mean?

[zaza]

Draw the tree!


After 2 rounds, the probability of A winning is 1/4, the probability of B winning is 0.25, otherwise C won the second game. After 3 rounds, the probability that C beat A is 1/8, the probability that C beat B is 1/8, otherwise C lost the 3rd round.

So far, then, the probability that A won is 1/4, the probability that B won is 1/4, the probability that C won is 1/4 and the remainder is the probability that nobody has yet won.

But in this remainder, we are back to the situation where A plays B and we've already discussed that. So after 3 rounds the probabilities are even at 1/4 each. Hence after another 3 rounds, they must still have even probabilities, but it won't be of 1/4 each - it will be slightly more.


Hence if we keep playing for ever, we'll end up with 3 even possibilities out of a total of 1, = 1/3 each.


If you still don't believe me, sum the geometric progression by working out the probabilities:

1/4 + 1/16 + 1/64 + ... for A and B

and


2x(1/8 + 1/32+...) for C.


The sum of a GP is a/(1-r) where a is the first term and r is the ratio between the terms.



zaza

[Noona]

OKKK got it.
thanks a lot zaza.