If A and B are two events such that P(A) = 0.4., `P (barB) = 0.3, P(A uu B)=0.9`, then what is the value of `P( barA uu barB)` ?

Options

A. A. 0.2
B. B. 0.5
C. C. 0.6
D. D. 0.7

Answer

To find the value of \( P(\bar{A} \cup \bar{B}) \), we can use the following steps:

### Step 1: Identify the given probabilities
We are given:
- \( P(A) = 0.4 \)
- \( P(\bar{B}) = 0.3 \)
- \( P(A \cup B) = 0.9 \)

### Step 2: Calculate \( P(\bar{A}) \)
Using the complement rule:
\[ P(\bar{A}) = 1 - P(A) = 1 - 0.4 = 0.6 \]

### Step 3: Calculate \( P(B) \)
Since \( P(\bar{B}) = 0.3 \), we can find \( P(B) \):
\[ P(B) = 1 - P(\bar{B}) = 1 - 0.3 = 0.7 \]

### Step 4: Calculate \( P(\bar{A} \cap \bar{B}) \)
We can use the formula:
\[ P(\bar{A} \cap \bar{B}) = 1 - P(A \cup B) \]
Substituting the value we have:
\[ P(\bar{A} \cap \bar{B}) = 1 - 0.9 = 0.1 \]

### Step 5: Use the formula for \( P(\bar{A} \cup \bar{B}) \)
Now we can use the formula:
\[ P(\bar{A} \cup \bar{B}) = P(\bar{A}) + P(\bar{B}) - P(\bar{A} \cap \bar{B}) \]
Substituting the values we calculated:
\[ P(\bar{A} \cup \bar{B}) = 0.6 + 0.3 - 0.1 \]
Calculating this gives:
\[ P(\bar{A} \cup \bar{B}) = 0.9 - 0.1 = 0.8 \]

### Final Answer
Thus, the value of \( P(\bar{A} \cup \bar{B}) \) is \( 0.8 \).

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