Let S and T are two events difined on a sample space with probabilities
`P(S)=0.5,P(T)=069,P(S//T)=0.5`
The value of P(S and T) is

Options

A. `0.3450`
B. `0.2500`
C. `0.6900`
D. `0.350`

Explanation

To find the value of \( P(S \cap T) \) (the probability of both events S and T occurring), we can use the formula for conditional probability.

### Step-by-Step Solution:

1. **Understand the given probabilities**:
- \( P(S) = 0.5 \)
- \( P(T) = 0.69 \)
- \( P(S | T) = 0.5 \)

2. **Use the formula for conditional probability**:
The formula for conditional probability is given by:
\[ P(S | T) = \frac{P(S \cap T)}{P(T)} \]
This means that the probability of event S occurring given that event T has occurred is equal to the probability of both events S and T occurring divided by the probability of event T.

3. **Rearrange the formula to find \( P(S \cap T) \)**:
We can rearrange the formula to solve for \( P(S \cap T) \):
\[ P(S \cap T) = P(S | T) \times P(T) \]

4. **Substitute the known values**:
Now, substitute the values we have:
\[ P(S \cap T) = 0.5 \times 0.69 \]

5. **Calculate the result**:
Now, perform the multiplication:
\[ P(S \cap T) = 0.5 \times 0.69 = 0.345 \]

6. **Conclusion**:
Therefore, the value of \( P(S \cap T) \) is \( 0.345 \).

### Final Answer:
\[ P(S \cap T) = 0.345 \]