Question

The probability that A wins B is p. And the probability that B wins A is q.

If you win 2 more times than opponent, the game is over.

What is the the probability that A two more wins and the game end?

Answer 1

A person can win only after even number of games (2n).

Assumption:

Probability of A winning in first 2 games = p*p = p²

• Let the game lasts for 2n points
• This will happen only if we have a tie till (2n-2) points
• For a tie till (2n-2) points,with no one having won,we need (n-1) occurences of AB or BA (probabilities being p*q or q*p)
• This sort of pattern has probability 2*p*q

Finally,the probability of A winning in 2n games is (2_*p_*q)^(n-1)_*p_*p

Example:

Lets take an example of 6 games. So 2n = 6 and n=3

In first 2 games,one can be won by A and other by B and similarly for next 2 games and the final 2 by A only.

The patterns can be like :

• ABABAA
• BABAAA
• ABBAAA
• BAABAA

For 1st two games it can be AB or BA,so the probability will be p*q

Similarly for next two games it will be p*q

So for first 4 games it is 2*p*q.

and for last 2 games it will be p*p

So probability of winning in 6 games = 2pq*p²

Answer : The probability of winning will be:

• p² + p²_*(2pq) + p²_*(2pq)²+ p²​​​​​​​_*(2pq)^{3​​​​​​​} + .......

This makes a G.P. whose sum can be calculated easily.