Let A and B be independent events such that P(A) =0.6 and P(B ) =0.4.
find ` P (barA nn barB)`

Options

A. `0.76`
B. `0.54`
C. `0.36`
D. `0.24`

Answer

To find \( P(\bar{A} \cap \bar{B}) \), where \( A \) and \( B \) are independent events with given probabilities \( P(A) = 0.6 \) and \( P(B) = 0.4 \), we will follow these steps:

### Step 1: Understand the Complement of Events
The complement of an event \( A \), denoted as \( \bar{A} \), is the event that \( A \) does not occur. Similarly, \( \bar{B} \) is the event that \( B \) does not occur.

### Step 2: Calculate the Probabilities of the Complements
To find \( P(\bar{A}) \) and \( P(\bar{B}) \), we use the formula:
\[ P(\bar{A}) = 1 - P(A) \]
\[ P(\bar{B}) = 1 - P(B) \]

Substituting the given values:
\[ P(\bar{A}) = 1 - 0.6 = 0.4 \]
\[ P(\bar{B}) = 1 - 0.4 = 0.6 \]

### Step 3: Use the Independence of Events
Since \( A \) and \( B \) are independent events, their complements \( \bar{A} \) and \( \bar{B} \) are also independent. Therefore, we can find the probability of their intersection using the multiplication rule for independent events:
\[ P(\bar{A} \cap \bar{B}) = P(\bar{A}) \times P(\bar{B}) \]

### Step 4: Substitute the Values
Now we substitute the values we calculated:
\[ P(\bar{A} \cap \bar{B}) = P(\bar{A}) \times P(\bar{B}) = 0.4 \times 0.6 \]

### Step 5: Calculate the Final Probability
Now, we calculate the product:
\[ P(\bar{A} \cap \bar{B}) = 0.4 \times 0.6 = 0.24 \]

### Conclusion
Thus, the probability \( P(\bar{A} \cap \bar{B}) \) is \( 0.24 \).

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