The probability of a problem being solved by two students are 1/2, 1/3. the proba-bility of the problem being solved is

Options

A. `2//3`
B. `4//3`
C. `1//3`
D. `1`

Answer

To solve the problem, we need to find the probability that at least one of the two students solves the problem. Let's denote the events as follows:

- Let \( A \) be the event that the first student solves the problem.
- Let \( B \) be the event that the second student solves the problem.

Given:
- The probability that the first student solves the problem, \( P(A) = \frac{1}{2} \).
- The probability that the second student solves the problem, \( P(B) = \frac{1}{3} \).

We want to find the probability that the problem is solved by at least one of the two students, which can be calculated using the formula:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Where \( P(A \cap B) \) is the probability that both students solve the problem.

### Step 1: Calculate \( P(A \cap B) \)
Since the events are independent, we can find \( P(A \cap B) \) as follows:

\[ P(A \cap B) = P(A) \times P(B) = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \]

### Step 2: Calculate \( P(A \cup B) \)
Now we can substitute the values into the formula:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Substituting the known values:

\[ P(A \cup B) = \frac{1}{2} + \frac{1}{3} - \frac{1}{6} \]

### Step 3: Find a common denominator
The least common multiple of 2, 3, and 6 is 6. We convert each fraction:

\[ P(A) = \frac{1}{2} = \frac{3}{6}, \quad P(B) = \frac{1}{3} = \frac{2}{6}, \quad P(A \cap B) = \frac{1}{6} \]

### Step 4: Substitute and simplify
Now substituting these values into the equation:

\[ P(A \cup B) = \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{3 + 2 - 1}{6} = \frac{4}{6} = \frac{2}{3} \]

### Final Answer
Thus, the probability that the problem is solved by at least one of the students is:

\[ \boxed{\frac{2}{3}} \]