Sixteen teams (including your favorite) enter a single-elimination tournament. Assume that your favorite team has a probability of winning game 1 of 0.9, and probabilities of winning its successive games (if it gets to play them) of 0.8, 0.7, and 0.6 respectively. Find the probability that your team is eliminated in the first round? In the second round? In the third round? The team wins the tournament?
- My professor skipped this section in class because he ran out of time and I am not sure if I understand the question correctly but am I supposed to start with the probability of 0.9 and subtract 0.1 from the previous round each time in order to find the rest of the probabilities? I am a little confused and any help would be much appreciated.
Best Answer
For the first round, since your team has a $0.9$ probability of winning, they have $0.1$ probability of being eliminated.
From the second round onwards, your team must win the previous rounds in order to progress (or, pessimistically, have a chance to be eliminated in subsequent rounds)
Your team has a $0.9$ probability of winning the first round, and a $1-0.8=0.2$ probability of losing the second. Hence the probability of your team being eliminated in the second round is $0.9 \times 0.2 = 0.18$. Similarly, they have a $0.9\times 0.8=0.72$ probability of progressing.
Using this logic, the remaining probabilities are (check them!):
Eliminated in the third round: $0.9\times 0.8\times 0.3=0.216$
Eliminated in the finals: $0.9\times 0.8\times 0.7\times 0.4=0.2016$
Winning the tournament: $0.9\times 0.8\times 0.7\times 0.6=0.3024$