# Thread: Probability question

1. ## Probability question

A and B play a series of similar game. The probability that A will win a set is 3/5, and there is no draw for each set. The game will continue until one of them, A or B win 2 sets. Find the probability that A wins the game.

I can't find any similar example from my reference book, and i also tried searching the internet for similar example but in vain... I attempted this question for like 3hours but also can't get the correct answer... 81/125

Can anyone help me?

2. ## Re: Probability question

Originally Posted by MichaelLight
A and B play a series of similar game. The probability that A will win a set is 3/5, and there is no draw for each set. The game will continue until one of them, A or B win 2 sets. Find the probability that A wins the game.

I can't find any similar example from my reference book, and i also tried searching the internet for similar example but in vain... I attempted this question for like 3hours but also can't get the correct answer... 81/125

Can anyone help me?
First, note that there can only ever be at most three rounds, because after those three rounds, someone will have won twice.

I'm going to write these possibilities as "winner for each round" (and N for not played)

AAN
ABA
ABB
BAA
BAB
BBN

Which of these possibilities would have A winning the game? What is the probability of each of those possibilities? What would the probability therefore be of A winning the game?

3. ## Re: Probability question

Thousands thanks for your help! I finally managed to get the answer now!

4. ## Re: Probability question

I just happened to run into a similar question. MichaelLight, is it possible that "The game will continue until one of them, A or B win 2 sets" actually implies that either A or B has to win two consecutive rounds in a row? (in this scenario, the game could obviously last more than 3 rounds)

5. ## Re: Probability question

Originally Posted by Maxim
I just happened to run into a similar question. MichaelLight, is it possible that "The game will continue until one of them, A or B win 2 sets" actually implies that either A or B has to win two consecutive rounds in a row? (in this scenario, the game could obviously last more than 3 rounds)
If that is the case, I doubt the answer would end up being 81/125...

6. ## Re: Probability question

Originally Posted by Prove It
If that is the case, I doubt the answer would end up being 81/125...
Yes, I had this approach:

"A1" being the situation where A has won one game (and maybe others before, but not more than one on a row), etc. And in MichaelLight's case, p=3/5.

Defining $\mathbb{P}_{A1}(A)$ as the probability that A wins from situation A1, and $\mathbb{P}_{B1}(A)$ the probability that A wins from situation B1:

$\mathbb{P}_{A1}(A) = p + (1-p)\mathbb{P}_{B1}(A)$

$\mathbb{P}_{B1}(A) = p\mathbb{P}_{A1}(A)$

$\Rightarrow \mathbb{P}_{A1}(A) = \frac{p}{1-p+p^2}\ ,\ \mathbb{P}_{B1}(A) = \frac{p^2}{1-p+p^2}$

Now at the start of the game, the probability that A wins is: $\mathbb{P}_{\text{start}}(A) = p\mathbb{P}_{A1}(A)+(1-p)\mathbb{P}_{B1}(A) = \frac{2p^2-p^3}{1-p+p^2}$

(for p=3/5 this is 63/95)