The probability of an event A is `(4)/(5)`. The probability of an event B, given that the event A occurs is `(1)/(5)`. The probability of event A, given that the event B occurs is `(2)/(3)`. The probability that neigher of the events occurs is

Options

A. `(3)/(25)`
B. `(2)/(5)`
C. `(1)/(25)`
D. `(2)/(15)`

Answer

To solve the problem, we need to find the probability that neither event A nor event B occurs. We will use the given probabilities and the formula for the probability of the union of two events.

### Step-by-step Solution:

1. **Identify Given Probabilities:**
- Probability of event A: \( P(A) = \frac{4}{5} \)
- Probability of event B given A: \( P(B|A) = \frac{1}{5} \)
- Probability of event A given B: \( P(A|B) = \frac{2}{3} \)

2. **Find Probability of Event B:**
We can use the formula for conditional probability:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]
Rearranging gives:
\[ P(A \cap B) = P(B|A) \cdot P(A) \]
Substituting the known values:
\[ P(A \cap B) = \frac{1}{5} \cdot \frac{4}{5} = \frac{4}{25} \]

3. **Find Probability of Event B Using \( P(A|B) \):**
Similarly, we can use:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Rearranging gives:
\[ P(B) = \frac{P(A \cap B)}{P(A|B)} \]
Substituting the known values:
\[ P(B) = \frac{\frac{4}{25}}{\frac{2}{3}} = \frac{4}{25} \cdot \frac{3}{2} = \frac{12}{50} = \frac{6}{25} \]

4. **Calculate Probability of A Union B:**
We use the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the values we found:
\[ P(A \cup B) = \frac{4}{5} + \frac{6}{25} - \frac{4}{25} \]
First, convert \( \frac{4}{5} \) to a fraction with a denominator of 25:
\[ \frac{4}{5} = \frac{20}{25} \]
Now substitute:
\[ P(A \cup B) = \frac{20}{25} + \frac{6}{25} - \frac{4}{25} = \frac{20 + 6 - 4}{25} = \frac{22}{25} \]

5. **Find Probability of Neither A Nor B:**
The probability that neither event A nor event B occurs is given by:
\[ P(A^c \cap B^c) = 1 - P(A \cup B) \]
Thus:
\[ P(A^c \cap B^c) = 1 - \frac{22}{25} = \frac{3}{25} \]

### Final Answer:
The probability that neither of the events occurs is \( \frac{3}{25} \).