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In a knockout tournament 16 equally skilled players namely P1, P2, -------- P16 are participating. In each round players are divided in pairs at random and winner from each pair moves in the next round. If P2 reaches the semifinal, then the probability that P1 will win the tournament is.
- a)3/64
- b)1/16
- c)1/20
- d)1/15
Correct answer is option 'C'. Can you explain this answer?
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In a knockout tournament 16 equally skilled players namely P1, P2, ---...
Let E1 = P1 win the tournament, E2 = P2 reaches the semifinal since all players are equally skilled and there are 4 persons in the semifinal 
both are in semifinal and P1 wins in semifinal and final
both are in semifinal and P1 wins in semifinal and final Most Upvoted Answer
In a knockout tournament 16 equally skilled players namely P1, P2, ---...
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In a knockout tournament 16 equally skilled players namely P1, P2, ---...
Problem Analysis:
In a knockout tournament, there are 16 players participating. In each round, players are divided into pairs at random and the winner from each pair moves on to the next round. We need to find the probability that P1 will win the tournament given that P2 reaches the semifinal.
Solution:
To solve this problem, we can use the concept of conditional probability. Let's break down the problem into smaller steps:
Step 1: Calculating the total number of possible outcomes:
In the first round, there are 16 players, so the total number of possible outcomes is 16C2 (combination of 16 players taken 2 at a time). Similarly, in the second round, there will be 8 players, so the total number of possible outcomes is 8C2. Following this, in the third round, there will be 4 players, so the total number of possible outcomes is 4C2. Finally, in the semifinal, there will be 2 players, so the total number of possible outcomes is 2C2 (which is 1). Therefore, the total number of possible outcomes can be calculated as:
16C2 * 8C2 * 4C2 * 1 = 16! / (2! * 2! * 2! * 1!) = 16! / (2^3) = 16! / 8
Step 2: Calculating the number of favorable outcomes:
Given that P2 reaches the semifinal, we need to calculate the number of favorable outcomes where P1 wins the tournament. Since P1 and P2 are on opposite sides of the bracket, they can only meet in the final. Therefore, P1 needs to win all the rounds after the semifinal in order to win the tournament. So, the number of favorable outcomes can be calculated as:
1 * 1 * 1 * 1 = 1
Step 3: Calculating the probability:
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. Therefore, the probability that P1 will win the tournament given that P2 reaches the semifinal can be calculated as:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 1 / (16! / 8)
Simplifying further:
Probability = 8 / 16!
Probability = 8 / (16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
Probability = 8 / 20922789888000
Probability = 1 / 2615348736000
Therefore, the probability that P1 will win the tournament given that P2 reaches the semifinal is 1/2615348736000, which is approximately equal to 1/20. Hence, the correct answer is option C.
In a knockout tournament, there are 16 players participating. In each round, players are divided into pairs at random and the winner from each pair moves on to the next round. We need to find the probability that P1 will win the tournament given that P2 reaches the semifinal.
Solution:
To solve this problem, we can use the concept of conditional probability. Let's break down the problem into smaller steps:
Step 1: Calculating the total number of possible outcomes:
In the first round, there are 16 players, so the total number of possible outcomes is 16C2 (combination of 16 players taken 2 at a time). Similarly, in the second round, there will be 8 players, so the total number of possible outcomes is 8C2. Following this, in the third round, there will be 4 players, so the total number of possible outcomes is 4C2. Finally, in the semifinal, there will be 2 players, so the total number of possible outcomes is 2C2 (which is 1). Therefore, the total number of possible outcomes can be calculated as:
16C2 * 8C2 * 4C2 * 1 = 16! / (2! * 2! * 2! * 1!) = 16! / (2^3) = 16! / 8
Step 2: Calculating the number of favorable outcomes:
Given that P2 reaches the semifinal, we need to calculate the number of favorable outcomes where P1 wins the tournament. Since P1 and P2 are on opposite sides of the bracket, they can only meet in the final. Therefore, P1 needs to win all the rounds after the semifinal in order to win the tournament. So, the number of favorable outcomes can be calculated as:
1 * 1 * 1 * 1 = 1
Step 3: Calculating the probability:
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. Therefore, the probability that P1 will win the tournament given that P2 reaches the semifinal can be calculated as:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 1 / (16! / 8)
Simplifying further:
Probability = 8 / 16!
Probability = 8 / (16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
Probability = 8 / 20922789888000
Probability = 1 / 2615348736000
Therefore, the probability that P1 will win the tournament given that P2 reaches the semifinal is 1/2615348736000, which is approximately equal to 1/20. Hence, the correct answer is option C.
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In a knockout tournament 16 equally skilled players namely P1, P2, -------- P16 are participating. In each round players are divided in pairs at random and winner from each pair moves in the next round. If P2 reaches the semifinal, then the probability that P1 will win the tournament is.a)3/64b)1/16c)1/20d)1/15Correct answer is option 'C'. Can you explain this answer?
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