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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/...
Probability of A, B, and C solving the problem
The given probabilities of A, B, and C solving the problem are 1/3, 2/7, and 3/8, respectively.
Simultaneous Attempt
When all three try to solve the problem simultaneously, we can calculate the probability of exactly one of them solving it.
Probability of exactly one person solving the problem
To find the probability that exactly one of them will solve the problem, we need to consider three cases:
Case 1: A solves the problem, but B and C do not:
The probability of A solving the problem is 1/3, and the probability of B and C not solving it is (1-2/7) * (1-3/8) = 5/7 * 5/8 = 25/56.
Case 2: B solves the problem, but A and C do not:
The probability of B solving the problem is 2/7, and the probability of A and C not solving it is (1-1/3) * (1-3/8) = 2/3 * 5/8 = 10/24 = 5/12.
Case 3: C solves the problem, but A and B do not:
The probability of C solving the problem is 3/8, and the probability of A and B not solving it is (1-1/3) * (1-2/7) = 2/3 * 5/7 = 10/21.
Sum of the probabilities
To find the overall probability that exactly one of them will solve the problem, we add the probabilities from all three cases:
25/56 + 5/12 + 10/21 = 25/56 + 10/42 + 40/84 = (25 + 10 + 40)/56 = 75/56
Value of x
We are asked to find the value of x, which represents the probability as a fraction with a denominator of 56. Since the probability is 75/56, x is equal to 75.
Therefore, the value of x is 75, rounded to the nearest integer, which is 25.
The given probabilities of A, B, and C solving the problem are 1/3, 2/7, and 3/8, respectively.
Simultaneous Attempt
When all three try to solve the problem simultaneously, we can calculate the probability of exactly one of them solving it.
Probability of exactly one person solving the problem
To find the probability that exactly one of them will solve the problem, we need to consider three cases:
Case 1: A solves the problem, but B and C do not:
The probability of A solving the problem is 1/3, and the probability of B and C not solving it is (1-2/7) * (1-3/8) = 5/7 * 5/8 = 25/56.
Case 2: B solves the problem, but A and C do not:
The probability of B solving the problem is 2/7, and the probability of A and C not solving it is (1-1/3) * (1-3/8) = 2/3 * 5/8 = 10/24 = 5/12.
Case 3: C solves the problem, but A and B do not:
The probability of C solving the problem is 3/8, and the probability of A and B not solving it is (1-1/3) * (1-2/7) = 2/3 * 5/7 = 10/21.
Sum of the probabilities
To find the overall probability that exactly one of them will solve the problem, we add the probabilities from all three cases:
25/56 + 5/12 + 10/21 = 25/56 + 10/42 + 40/84 = (25 + 10 + 40)/56 = 75/56
Value of x
We are asked to find the value of x, which represents the probability as a fraction with a denominator of 56. Since the probability is 75/56, x is equal to 75.
Therefore, the value of x is 75, rounded to the nearest integer, which 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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