To solve the problem, we need to calculate the probabilities based on the chances of each student solving the problem. Let's denote the students as A, B, and C, with their respective probabilities of solving the problem as follows:
- Student A: P(A) = 1/2
- Student B: P(B) = 1/3
- Student C: P(C) = 1/4
Probability that the problem is solved
The probability that at least one student solves the problem can be found by calculating the complement of the probability that none of them solves it.
The probability that each student does not solve the problem is:
- P(A does not solve) = 1 - P(A) = 1/2
- P(B does not solve) = 1 - P(B) = 2/3
- P(C does not solve) = 1 - P(C) = 3/4
Now, we multiply these probabilities together:
P(none solve) = (1/2) * (2/3) * (3/4) = 1/4
Thus, the probability that at least one solves the problem is:
P(at least one solves) = 1 - P(none solve) = 1 - 1/4 = 3/4
Probability that exactly one student solves it
To find the probability that exactly one student solves the problem, we need to consider each student solving it while the others do not:
- Only A solves: P(A) * P(B does not solve) * P(C does not solve) = (1/2) * (2/3) * (3/4) = 1/4
- Only B solves: P(B) * P(A does not solve) * P(C does not solve) = (1/3) * (1/2) * (3/4) = 1/8
- Only C solves: P(C) * P(A does not solve) * P(B does not solve) = (1/4) * (1/2) * (2/3) = 1/12
Now, we sum these probabilities:
P(exactly one solves) = 1/4 + 1/8 + 1/12
To add these fractions, we find a common denominator, which is 24:
- 1/4 = 6/24
- 1/8 = 3/24
- 1/12 = 2/24
Thus, P(exactly one solves) = 6/24 + 3/24 + 2/24 = 11/24
At least one student may solve it
This is the same as the first calculation we did. The probability that at least one student solves the problem is:
P(at least one solves) = 3/4
Summary of Results
- Probability that the problem is solved: 3/4
- Probability that exactly one solves it: 11/24
- Probability that at least one may solve it: 3/4
