Given the probability that A can solve a problem is 2/3, and the probability that B can solve the same problem is 3/5, find the probability that (i) at least one of A and B will solve the problem (ii) None of the two will solve the problem.

Explanation

To solve the problem, we need to find the probabilities for two scenarios involving two individuals, A and B, who can solve a problem independently.

### Given:
- Probability that A can solve the problem, \( P(A) = \frac{2}{3} \)
- Probability that B can solve the problem, \( P(B) = \frac{3}{5} \)

### Step 1: Find the probability that A does not solve the problem.
The probability that A does not solve the problem is the complement of the probability that A solves it:
\[ P(A') = 1 - P(A) = 1 - \frac{2}{3} = \frac{1}{3} \]

### Step 2: Find the probability that B does not solve the problem.
Similarly, the probability that B does not solve the problem is:
\[ P(B') = 1 - P(B) = 1 - \frac{3}{5} = \frac{2}{5} \]

### Step 3: Find the probability that neither A nor B solves the problem.
Since A and B are independent, the probability that neither A nor B solves the problem is the product of their individual probabilities of not solving the problem:
\[ P(A' \cap B') = P(A') \times P(B') = \frac{1}{3} \times \frac{2}{5} = \frac{2}{15} \]

### Step 4: Find the probability that at least one of A or B solves the problem.
The probability that at least one of A or B solves the problem can be found using the complement of the probability that neither A nor B solves the problem:
\[ P(A \cup B) = 1 - P(A' \cap B') = 1 - \frac{2}{15} = \frac{15}{15} - \frac{2}{15} = \frac{13}{15} \]

### Final Answers:
1. The probability that at least one of A and B will solve the problem is \( \frac{13}{15} \).
2. The probability that none of the two will solve the problem is \( \frac{2}{15} \).