Daily Math Puzzle: 2026-07-23
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2026-07-23
In a small, single-elimination sports tournament, there are four teams: Alpha, Beta, Gamma, and Delta. The tournament structure is as follows: Alpha plays Beta in the first semi-final, and Gamma plays Delta in the second semi-final. The winners of these matches proceed to the final. Historically, Team Alpha has a 70% chance of winning any match it plays. Team Beta, being weaker, has only a 30% chance of winning any match it plays. Team Gamma and Team Delta are equally matched, meaning each has a 50% chance of winning against the other. A curious phenomenon is observed: if Team Gamma manages to reach the final, they inexplicably gain a significant performance boost, giving them an 80% chance of winning the final, *regardless* of whether they play Alpha or Beta. What is the probability that Team Gamma wins the entire tournament?
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Solution
40% — For Team Gamma to win the tournament, two conditions must be met:
1. Team Gamma must win its semi-final match.
2. Team Gamma must win the final match.
First, let's calculate the probability of Team Gamma winning its semi-final. Team Gamma plays Team Delta, and they are equally matched. So, P(Gamma wins semi-final) = 50% = 0.5.
Next, if Team Gamma wins its semi-final, it reaches the final. The puzzle states that 'if Team Gamma manages to reach the final, they inexplicably gain a significant performance boost, giving them an 80% chance of winning the final, *regardless* of whether they play Alpha or Beta.' This is the key piece of information.
This means that the probability of Team Gamma winning the final, *given that they have reached it*, is explicitly 80% or 0.8. The information about Alpha's and Beta's individual probabilities of winning their semi-final or their general match-winning chances against each other is irrelevant for Gamma's final victory probability because the 80% figure applies universally once Gamma is in the final.
Therefore, the probability that Team Gamma wins the entire tournament is the product of the probability of them winning their semi-final AND the probability of them winning the final (given they reached it):
P(Gamma wins tournament) = P(Gamma wins semi-final) * P(Gamma wins final | Gamma reaches final)
P(Gamma wins tournament) = 0.5 * 0.8 = 0.40.
So, the probability that Team Gamma wins the entire tournament is 40%.
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