Ques: Two players are competing in a tournament which consists of three matches. The probability of player1 winning the first match is 0.2, winning the second match is 0.5 and winning the third match is 0.6. The probability of player2 winning the first match is 0.8, winning the second match is 0.5 and winning the third match is 0.4.
A player wins the tournament is he wins all the matches. Otherwise, the tournament is played again. The tournament is played again and again until a player wins all matches and hence wins the tournament.
What is the probability that player1 wins the tournament?
My approach: Since the matches are independent of each other, the probability of player1 winning all the matches is 0.2*0.5*0.6. There are 7 other possible outcomes of tournaments:-
0.2*0.5*(1-0.6) i.e 0.2*0.5*0.4
0.2*(1-0.5)*0.6 i.e 0.2*0.5*0.4
… and so on.
I'm am doubtful as to how to progress from here. Any help would be appreciated.
Best Answer
I believe the simplest way is to ignore altogether rounds where neither win.
P(A wins in a round) = $0.2\cdot0.5\cdot0.6 = 0.06$
P(B wins in a round) = $0.8\cdot0.5\cdot0.4 = 0.16$
Odds in favour of A = 6/16 = 3/8, so
P(A wins) = 3/11
P(B wins) = 8/11