If 2P(A) =P(B) `=5/13 and P(A//B)=2/5` , then `P(A cup B)` is

Options

A. `15/26`
B. `11/26`
C. `15/39`
D. `11/39`

Answer

To solve the problem step by step, we will use the given information and apply the formulas of probability.

### Given:
1. \( 2P(A) = P(B) = \frac{5}{13} \)
2. \( P(A|B) = \frac{2}{5} \)

### Step 1: Find \( P(A) \)
From the first equation, we can express \( P(A) \):
\[ P(A) = \frac{P(B)}{2} = \frac{5/13}{2} = \frac{5}{26} \]

### Step 2: Find \( P(B) \)
From the given information, we already have:
\[ P(B) = \frac{5}{13} \]

### Step 3: Find \( P(A \cap B) \)
Using the conditional probability formula:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
We can rearrange this to find \( P(A \cap B) \):
\[ P(A \cap B) = P(A|B) \cdot P(B) \]
Substituting the known values:
\[ P(A \cap B) = \frac{2}{5} \cdot \frac{5}{13} = \frac{2 \cdot 5}{5 \cdot 13} = \frac{2}{13} \]

### Step 4: Find \( P(A \cup B) \)
Using the formula for the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the values we found:
\[ P(A \cup B) = \frac{5}{26} + \frac{5}{13} - \frac{2}{13} \]
To add these fractions, we need a common denominator. The least common multiple of 26 and 13 is 26.
Converting \( P(B) \) and \( P(A \cap B) \) to have a denominator of 26:
\[ P(B) = \frac{5}{13} = \frac{10}{26} \]
\[ P(A \cap B) = \frac{2}{13} = \frac{4}{26} \]
Now substituting these values back:
\[ P(A \cup B) = \frac{5}{26} + \frac{10}{26} - \frac{4}{26} = \frac{5 + 10 - 4}{26} = \frac{11}{26} \]

### Final Answer:
\[ P(A \cup B) = \frac{11}{26} \]

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