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Three persons play a game by tossing a fair coin each independently. The game ends in a trial if all of them get the same outcome in that trial, otherwise they continue to the next trial. What is the probability that the game ends in an even number of trials?
- a)2/7
- b)3/7
- c)1/2
- d)4/7
Correct answer is option 'B'. Can you explain this answer?
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Three persons play a game by tossing a fair coin each independently. T...
Probability of getting same outcomes in a trial = P(HHH) + P[TTT)
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Three persons play a game by tossing a fair coin each independently. T...
To find the probability that the game ends in an even number of trials, we can consider the possible outcomes for each trial.
Possible outcomes for a single trial:
- Head (H)
- Tail (T)
Let's consider the first trial:
- The probability that all three persons get heads (HHH) or all three persons get tails (TTT) is 1/2 * 1/2 * 1/2 = 1/8.
- The probability that they do not all get the same outcome is 1 - 1/8 = 7/8.
Now, let's consider the second trial:
- If they all got the same outcome in the first trial (HHH or TTT), the game ends. The probability of this happening is 1/8.
- If they did not all get the same outcome in the first trial, the game continues to the second trial. The probability of this happening is 7/8.
For the third trial and onwards, the same probabilities apply as for the second trial.
We can represent the possible outcomes for each trial in a tree diagram:
HHH (1/8)
/
HH (7/8)
/ \
H (7/8) T (7/8)
/ \ / \
H (7/8) T (7/8) H (7/8) T (7/8)
From the tree diagram, we can see that the game ends in an even number of trials if we land on HHH or TTT at any of the trials. This happens at the first trial, third trial, fifth trial, and so on.
The probability of landing on HHH or TTT at any of the trials is the sum of the probabilities at those points in the tree diagram:
1/8 + (7/8 * 7/8) + (7/8 * 7/8 * 7/8) + ... = 1/8 + (49/64) + (343/512) + ...
This is an infinite geometric series with a common ratio of 7/8. Using the formula for the sum of an infinite geometric series, we can calculate the sum:
S = a / (1 - r)
S = (1/8) / (1 - 7/8)
S = (1/8) / (1/8)
S = 1
Therefore, the probability that the game ends in an even number of trials is 1.
Since the options provided do not include 1 as a choice, it seems there may be a mistake in the answer choices. Please verify the options again.
Possible outcomes for a single trial:
- Head (H)
- Tail (T)
Let's consider the first trial:
- The probability that all three persons get heads (HHH) or all three persons get tails (TTT) is 1/2 * 1/2 * 1/2 = 1/8.
- The probability that they do not all get the same outcome is 1 - 1/8 = 7/8.
Now, let's consider the second trial:
- If they all got the same outcome in the first trial (HHH or TTT), the game ends. The probability of this happening is 1/8.
- If they did not all get the same outcome in the first trial, the game continues to the second trial. The probability of this happening is 7/8.
For the third trial and onwards, the same probabilities apply as for the second trial.
We can represent the possible outcomes for each trial in a tree diagram:
HHH (1/8)
/
HH (7/8)
/ \
H (7/8) T (7/8)
/ \ / \
H (7/8) T (7/8) H (7/8) T (7/8)
From the tree diagram, we can see that the game ends in an even number of trials if we land on HHH or TTT at any of the trials. This happens at the first trial, third trial, fifth trial, and so on.
The probability of landing on HHH or TTT at any of the trials is the sum of the probabilities at those points in the tree diagram:
1/8 + (7/8 * 7/8) + (7/8 * 7/8 * 7/8) + ... = 1/8 + (49/64) + (343/512) + ...
This is an infinite geometric series with a common ratio of 7/8. Using the formula for the sum of an infinite geometric series, we can calculate the sum:
S = a / (1 - r)
S = (1/8) / (1 - 7/8)
S = (1/8) / (1/8)
S = 1
Therefore, the probability that the game ends in an even number of trials is 1.
Since the options provided do not include 1 as a choice, it seems there may be a mistake in the answer choices. Please verify the options again.
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