In a contest, Richard,Jennifer and Alex are asked to solve a problem. The respective probabilities that they solve the problem are 1/6,1/8 and 1/3 , Calculate the probability that
- None of them solves the problem
- At least one of them solves the problem
- Only one of them solves the problem
To calculate probabilities in this scenario, we can use basic probability rules. Let's calculate each probability step by step:
1. Probability that none of them solves the problem:
To calculate the probability that none of them solves the problem, we need to calculate the probability that each of them individually does not solve the problem, and then multiply those probabilities together.
Probability that Richard does not solve the problem = 1 - 1/6 = 5/6
Probability that Jennifer does not solve the problem = 1 - 1/8 = 7/8
Probability that Alex does not solve the problem = 1 - 1/3 = 2/3
Now, we multiply these probabilities together:
(5/6) * (7/8) * (2/3) = 35/144
So, the probability that none of them solves the problem is 35/144.
2. Probability that at least one of them solves the problem:
To calculate the probability that at least one of them solves the problem, we can use the complement rule. The complement of "at least one of them solves the problem" is "none of them solves the problem". So, we can subtract the probability that none of them solves the problem from 1.
Probability that at least one of them solves the problem = 1 - (35/144) = 109/144
So, the probability that at least one of them solves the problem is 109/144.
3. Probability that only one of them solves the problem:
To calculate the probability that only one of them solves the problem, we need to calculate the probability for each person individually and then sum them together.
Probability that only Richard solves the problem:
Probability that Richard solves the problem: 1/6
Probability that Jennifer does not solve the problem: 7/8
Probability that Alex does not solve the problem: 2/3
(1/6) * (7/8) * (2/3) = 7/144
Probability that only Jennifer solves the problem:
Probability that Richard does not solve the problem: 5/6
Probability that Jennifer solves the problem: 1/8
Probability that Alex does not solve the problem: 2/3
(5/6) * (1/8) * (2/3) = 5/144
Probability that only Alex solves the problem:
Probability that Richard does not solve the problem: 5/6
Probability that Jennifer does not solve the problem: 7/8
Probability that Alex solves the problem: 1/3
(5/6) * (7/8) * (1/3) = 35/144
Now, we add these probabilities together:
(7/144) + (5/144) + (35/144) = 47/144
So, the probability that only one of them solves the problem is 47/144.