Martina is playing in an 8 player tournament. She knows she will beat everyone except Ash, who will beat everyone. In the first round, players are organized randomly into 4 pairs and the winner of each match proceeds to the second round. In the second round there are 2 matches and the winners of these matches proceed to the final. What's the probability that Martina does not get to the final? The one way to model this random tournament is to have a sequence of 8 players like this, and then the first 2 players get paired up, the next 2 players get paired up, and so on like this. Now out of these 2 players and these 2 players, their winners will play in the second round and similarly for this pair. And finally the winners out of the big blue boxes will play each other in the final round. So now Ash has to go somewhere in this sequence and it doesn't really matter where he ends up so without loss of generality let's put him here. So in the first round Ash plays this guy right here. These 2 people also play each other and so in the second round Ash is going to be the winner out of this blue box and he's going to go to the final because he beats everyone. Now that means if Martina ends up in any of this blue box region, she's going to lose to Ash before the final. Otherwise if Martina ends up in any of these regions, she's going to get to the final because she beats everyone in here. Martina gets into each of these empty spots with equal probability and there are 3 spots where she doesn't make the finals and 4 spots where she does make the finals, so the final probability is 3 out of 7.

tournament

Martina

Ash

semifinals

probability