in a contest, Bello. Adamu, and Lawi are asked to solve a mathematical problem. the probability that they solve the problem correctly are 1/5, 2/3, and 2/5. find the probability that:(1)None of them solve the problem correctly(2)At least one of them solve the problem correctly(3)Only one of them solves the problem correctly

Prob(all wrong) = (4/5)(1/3)(3/5)

= 12/75 = 4/25

prob(at least one) = 1 - 4/25 = 21/25

prob(only one correct)
---> RWW, WRW, WWR
= (1/5)((1/3)(3/5) + (4/5)(2/3)(3/5) + (4/5)(1/3(2/5)
= 3/75 + 24/75 + 8/75
= 35/75
= 7/15

check my arithmetic

To find the probability for each case, we need to consider the individual probabilities of each person not solving the problem correctly, solving it correctly, or only one person solving it correctly. Let's calculate each case step by step:

(1) None of them solve the problem correctly:
To find the probability that none of them solve the problem correctly, we multiply their individual probabilities together.
Probability of Bello not solving the problem correctly: 1 - 1/5 = 4/5
Probability of Adamu not solving the problem correctly: 1 - 2/3 = 1/3
Probability of Lawi not solving the problem correctly: 1 - 2/5 = 3/5

Now we multiply these individual probabilities together:
Probability of none of them solving the problem correctly = (4/5) * (1/3) * (3/5) = 12/75 = 4/25

(2) At least one of them solves the problem correctly:
To find the probability that at least one person solves the problem correctly, we need to calculate the complement of none of them solving it correctly. The complement probability is 1 minus the probability of none of them solving it correctly.
Probability of at least one person solving the problem correctly = 1 - probability of none of them solving it correctly
Probability of at least one person solving the problem correctly = 1 - 4/25 = 21/25

(3) Only one of them solves the problem correctly:
To find the probability that only one person solves the problem correctly, we need to calculate three separate cases: Bello solves it, Adamu solves it, and Lawi solves it. We will then add these probabilities together.
Probability that Bello solves it but the other two don't:
Probability of Bello solving the problem correctly = 1/5
Probability of Adamu not solving the problem correctly = 1 - 2/3 = 1/3
Probability of Lawi not solving the problem correctly = 1 - 2/5 = 3/5
Probability of Bello solving it alone: (1/5) * (1/3) * (3/5) = 3/75

Probability that Adamu solves it but the other two don't:
Probability of Bello not solving the problem correctly = 1 - 1/5 = 4/5
Probability of Adamu solving the problem correctly = 2/3
Probability of Lawi not solving the problem correctly = 1 - 2/5 = 3/5
Probability of Adamu solving it alone: (4/5) * (2/3) * (3/5) = 24/75 = 8/25

Probability that Lawi solves it but the other two don't:
Probability of Bello not solving the problem correctly = 1 - 1/5 = 4/5
Probability of Adamu not solving the problem correctly = 1 - 2/3 = 1/3
Probability of Lawi solving the problem correctly = 2/5
Probability of Lawi solving it alone: (4/5) * (1/3) * (2/5) = 8/75

Now we add these three probabilities together:
Probability that only one person solves the problem correctly = (3/75) + (8/25) + (8/75) = 27/75 = 9/25

So, the probabilities are:
(1) Probability that none of them solve the problem correctly = 4/25
(2) Probability that at least one person solves the problem correctly = 21/25
(3) Probability that only one person solves the problem correctly = 9/25