If P(not B) = 0.65, `P(A cup B)=0.85` and A and B are independent events, then find P(A).

Answer

To solve the problem, we will follow these steps:

**Step 1: Find P(B)**
Given that P(not B) = 0.65, we can find P(B) using the formula:
\[ P(B) = 1 - P(\text{not } B) \]
Substituting the given value:
\[ P(B) = 1 - 0.65 = 0.35 \]

**Step 2: Use the formula for P(A ∪ B)**
We know that for any two events A and B:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Since A and B are independent events, we have:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Thus, we can rewrite the formula as:
\[ P(A \cup B) = P(A) + P(B) - P(A) \cdot P(B) \]

**Step 3: Substitute known values**
We know:
- P(A ∪ B) = 0.85
- P(B) = 0.35

Substituting these into the equation:
\[ 0.85 = P(A) + 0.35 - P(A) \cdot 0.35 \]

**Step 4: Rearrange the equation**
Let's rearrange the equation to isolate P(A):
\[ 0.85 = P(A) + 0.35 - 0.35 P(A) \]
Combine like terms:
\[ 0.85 = P(A)(1 - 0.35) + 0.35 \]
\[ 0.85 = P(A)(0.65) + 0.35 \]

**Step 5: Solve for P(A)**
Now, isolate P(A):
\[ 0.85 - 0.35 = 0.65 P(A) \]
\[ 0.50 = 0.65 P(A) \]
Now divide both sides by 0.65:
\[ P(A) = \frac{0.50}{0.65} \]

**Step 6: Calculate P(A)**
Calculating the value:
\[ P(A) \approx 0.7692 \]

Thus, the final answer is:
\[ P(A) \approx 0.7692 \]

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