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 find the probability that neither of the events A nor B occurs, we can follow these steps:

### Step 1: Understand the given probabilities
- The probability of event A, \( P(A) = \frac{4}{5} \)
- The probability of event B given that event A occurs, \( P(B|A) = \frac{1}{5} \)
- The probability of event A given that event B occurs, \( P(A|B) = \frac{2}{3} \)

### Step 2: Use the formula for the probability of the union of two events
We want to find the probability that neither A nor B occurs, which can be expressed as:
\[ P(A' \cap B') = 1 - P(A \cup B) \]
where \( A' \) and \( B' \) are the complements of events A and B, respectively.

### Step 3: Calculate \( P(A \cup B) \)
Using the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

### Step 4: Find \( P(A \cap B) \)
We can find \( P(A \cap B) \) using the conditional probability:
\[ P(A \cap B) = P(B|A) \cdot P(A) \]
Substituting the known values:
\[ P(A \cap B) = \frac{1}{5} \cdot \frac{4}{5} = \frac{4}{25} \]

### Step 5: Find \( P(B) \)
We can also find \( P(B) \) using the conditional probability:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Rearranging gives:
\[ P(B) = \frac{P(A \cap B)}{P(A|B)} \]
Substituting the known values:
\[ P(B) = \frac{\frac{4}{25}}{\frac{2}{3}} = \frac{4}{25} \cdot \frac{3}{2} = \frac{6}{25} \]

### Step 6: Substitute values into the union formula
Now we can substitute \( P(A) \), \( P(B) \), and \( P(A \cap B) \) into the union formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
\[ P(A \cup B) = \frac{4}{5} + \frac{6}{25} - \frac{4}{25} \]
To add these fractions, convert \( \frac{4}{5} \) to a fraction with a denominator of 25:
\[ P(A \cup B) = \frac{20}{25} + \frac{6}{25} - \frac{4}{25} = \frac{20 + 6 - 4}{25} = \frac{22}{25} \]

### Step 7: Calculate \( P(A' \cap B') \)
Now we can find the probability that neither A nor B occurs:
\[ P(A' \cap B') = 1 - P(A \cup B) = 1 - \frac{22}{25} = \frac{3}{25} \]

### Final Answer
The probability that neither of the events occurs is:
\[ \boxed{\frac{3}{25}} \]