# Probability using MasterChef Competition

There is a red team and a blue team. The members of the red team are A,B, C. The members of the blue team: D,E,F. One of this team will win a given challenge. One of the member of the losing team who performs the worst will be eliminated. What is the probability that both B and E progress to the next round.

To determine the probability that both B and E progress to the next round, we need to consider the following scenarios:

- Red Team wins, and E is not the worst performer on the Blue Team.
- Blue Team wins, and B is not the worst performer on the Red Team.

Firstly, the team that wins (Red or Blue) has no member eliminated. That means if Red Team wins, B progresses for sure. Similarly, if Blue Team wins, E progresses.

Now let’s consider each scenario:

### Scenario 1: Red Team wins

- For E to progress, either D or F must perform worse than E. The probability that E is not the worst performer on the Blue Team is 2/3.

### Scenario 2: Blue Team wins

- For B to progress, either A or C must perform worse than B. The probability that B is not the worst performer on the Red Team is 2/3.

Now, assuming that each team has an equal chance of winning, the probability of either team winning is 1/2.

Therefore, the overall probability that both B and E progress to the next round would be:

- Probability(Red Team wins and E is not worst) = (1/2) * (2/3) = 1/3
- Probability(Blue Team wins and B is not worst) = (1/2) * (2/3) = 1/3

Either of these events happening would mean that both B and E progress, so we add the probabilities:

1/3 + 1/3 = 2/3

So, the probability that both B and E progress to the next round is 2/3.

## Claude Answer

- There are 2 teams - red (A,B,C) and blue (D,E,F)
- One team will win, the other will lose
- The worst performer from the losing team will be eliminated
- We want probability that B and E both progress

To calculate this:

- There are 2 scenarios where B and E both progress:
- Red team loses, A performs worst
- Blue team loses, D performs worst

- There are 6 total scenarios for which team can lose and worst performer (Winning team members automatically progress)
- So probability B and E progress = 2/6 = 1/3

To summarize:

- B and E progress if their team wins or they are not worst performer on losing team
- This occurs in 2 of the 6 total scenarios
- So the probability is 1/3