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The probabilities of A, B and C solving a problem are 1/3, 2/7 and 3/8, respectively. If all three try to solve the problem simultaneously, then the probability that exactly one of them will solve it is x/56. Find the value of x. (Answer up to the nearest integer)
Correct answer is '25'. Can you explain this answer?
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The probabilities of A, B and C solving a problem are 1/3, 2/7 and 3/...
Understanding the Problem
To find the probability that exactly one of A, B, or C solves the problem, we need to first determine the probabilities of each individual solving the problem and not solving it.
Given Probabilities
- Probability that A solves: P(A) = 1/3
- Probability that B solves: P(B) = 2/7
- Probability that C solves: P(C) = 3/8
Calculating Not Solving Probabilities
- Probability that A does not solve: P(A') = 1 - P(A) = 2/3
- Probability that B does not solve: P(B') = 1 - P(B) = 5/7
- Probability that C does not solve: P(C') = 1 - P(C) = 5/8
Finding Probabilities for Exactly One Solver
To find the probability that exactly one of them solves the problem, we analyze three scenarios:
1. Only A solves:
Probability = P(A) * P(B') * P(C')
= (1/3) * (5/7) * (5/8)
2. Only B solves:
Probability = P(A') * P(B) * P(C')
= (2/3) * (2/7) * (5/8)
3. Only C solves:
Probability = P(A') * P(B') * P(C)
= (2/3) * (5/7) * (3/8)
Calculating Each Scenario
- For A only:
(1/3) * (5/7) * (5/8) = 25/168
- For B only:
(2/3) * (2/7) * (5/8) = 20/168
- For C only:
(2/3) * (5/7) * (3/8) = 30/168
Total Probability for Exactly One Solver
Now, sum these probabilities:
Total = (25/168) + (20/168) + (30/168) = 75/168 = 25/56
Final Value of x
Since the result is expressed as x/56, we find:
x = 25
Thus, the value of x is 25.
To find the probability that exactly one of A, B, or C solves the problem, we need to first determine the probabilities of each individual solving the problem and not solving it.
Given Probabilities
- Probability that A solves: P(A) = 1/3
- Probability that B solves: P(B) = 2/7
- Probability that C solves: P(C) = 3/8
Calculating Not Solving Probabilities
- Probability that A does not solve: P(A') = 1 - P(A) = 2/3
- Probability that B does not solve: P(B') = 1 - P(B) = 5/7
- Probability that C does not solve: P(C') = 1 - P(C) = 5/8
Finding Probabilities for Exactly One Solver
To find the probability that exactly one of them solves the problem, we analyze three scenarios:
1. Only A solves:
Probability = P(A) * P(B') * P(C')
= (1/3) * (5/7) * (5/8)
2. Only B solves:
Probability = P(A') * P(B) * P(C')
= (2/3) * (2/7) * (5/8)
3. Only C solves:
Probability = P(A') * P(B') * P(C)
= (2/3) * (5/7) * (3/8)
Calculating Each Scenario
- For A only:
(1/3) * (5/7) * (5/8) = 25/168
- For B only:
(2/3) * (2/7) * (5/8) = 20/168
- For C only:
(2/3) * (5/7) * (3/8) = 30/168
Total Probability for Exactly One Solver
Now, sum these probabilities:
Total = (25/168) + (20/168) + (30/168) = 75/168 = 25/56
Final Value of x
Since the result is expressed as x/56, we find:
x = 25
Thus, the value of x is 25.
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Community Answer
The probabilities of A, B and C solving a problem are 1/3, 2/7 and 3/...
Let E1, E2 and E3 be the events that the problem is solved by A, B and C, respectively. Then,
P(E1) = 1/3, P(E2) = 2/7 and P(E3) = 3/8
Exactly one of A, B and C can solve the problem in the following mutually exclusive ways.
(i) A solves and B and C do not solve.
(ii) B solves and A and C do not solve.
(iii) C solves and A and B do not solve.
Probability that only one will solve the problem
= P(A) and P(not B) and P(not C) + P(not A) and P(B) and P(not C) + P(not A) and P(not B) and P(C)
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The probabilities of A, B and C solving a problem are 1/3, 2/7 and 3/8, respectively. If all three try to solve the problem simultaneously, then the probability that exactly one of them will solve it is x/56. Find the value of x. (Answer up to the nearest integer)Correct answer is '25'. Can you explain this answer?
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