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A problem of mathematics is given to three students A, B and C, and their respective probabilities of solving the problem is 1/2, 1/3 and 1/4. The probability that the problem is solved is
- a)3/4
- b)1/2
- c)2/3
- d)1/3
Correct answer is option 'A'. Can you explain this answer?
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A problem of mathematics is given to three students A, B and C, and th...
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A problem of mathematics is given to three students A, B and C, and th...
To find the probability that the problem is solved, we need to consider the probabilities of each student solving the problem and then calculate the probability that at least one of them solves it.
Let's denote the events as follows:
A: Student A solves the problem
B: Student B solves the problem
C: Student C solves the problem
The probability of A solving the problem is given as 1/2, which means that the probability of A not solving the problem is 1 - 1/2 = 1/2. Similarly, the probabilities for B and C not solving the problem are 2/3 and 3/4 respectively.
Now, we need to find the probability that at least one of the students solves the problem. This can be done by finding the probability of the complement event, i.e., the probability that none of the students solve the problem.
Using the multiplication principle, the probability that none of the students solve the problem is given by:
P(not A and not B and not C) = P(not A) * P(not B) * P(not C)
Substituting the respective probabilities:
P(not A and not B and not C) = (1/2) * (2/3) * (3/4) = 1/4
Now, we can find the probability that at least one of the students solves the problem by subtracting the probability of none of them solving it from 1:
P(at least one student solves the problem) = 1 - P(not A and not B and not C) = 1 - 1/4 = 3/4
Therefore, the probability that the problem is solved is 3/4, which corresponds to option A.
Let's denote the events as follows:
A: Student A solves the problem
B: Student B solves the problem
C: Student C solves the problem
The probability of A solving the problem is given as 1/2, which means that the probability of A not solving the problem is 1 - 1/2 = 1/2. Similarly, the probabilities for B and C not solving the problem are 2/3 and 3/4 respectively.
Now, we need to find the probability that at least one of the students solves the problem. This can be done by finding the probability of the complement event, i.e., the probability that none of the students solve the problem.
Using the multiplication principle, the probability that none of the students solve the problem is given by:
P(not A and not B and not C) = P(not A) * P(not B) * P(not C)
Substituting the respective probabilities:
P(not A and not B and not C) = (1/2) * (2/3) * (3/4) = 1/4
Now, we can find the probability that at least one of the students solves the problem by subtracting the probability of none of them solving it from 1:
P(at least one student solves the problem) = 1 - P(not A and not B and not C) = 1 - 1/4 = 3/4
Therefore, the probability that the problem is solved is 3/4, which corresponds to option A.
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