Let A and B be two events such that P(A) = 3/7, P(B) = 4/7 and P(A`uu`B) = 5/7, Then `(P(A|B))/(P(A'|B))` equals

Options

A. `1//3`
B. `2//3`
C. `1//2`
D. `3//8`

Answer

To solve the problem step by step, we will use the given probabilities and the properties of conditional probability.

### Step 1: Understand the Given Information
We are given:
- \( P(A) = \frac{3}{7} \)
- \( P(B) = \frac{4}{7} \)
- \( P(A' \cup B) = \frac{5}{7} \)

### Step 2: Use the Formula for Union of Events
We know that:
\[ P(A' \cup B) = P(A') + P(B) - P(A' \cap B) \]
We can also express \( P(A') \) as \( 1 - P(A) \):
\[ P(A') = 1 - P(A) = 1 - \frac{3}{7} = \frac{4}{7} \]

### Step 3: Substitute the Values into the Union Formula
Now substituting the values into the union formula:
\[ \frac{5}{7} = \frac{4}{7} + \frac{4}{7} - P(A' \cap B) \]
This simplifies to:
\[ \frac{5}{7} = \frac{8}{7} - P(A' \cap B) \]

### Step 4: Solve for \( P(A' \cap B) \)
Rearranging gives:
\[ P(A' \cap B) = \frac{8}{7} - \frac{5}{7} = \frac{3}{7} \]

### Step 5: Find \( P(A \cap B) \)
Now we can find \( P(A \cap B) \) using the formula for the union of events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
We know \( P(A \cup B) = 1 - P(A') = 1 - \frac{4}{7} = \frac{3}{7} \):
\[ \frac{3}{7} = \frac{3}{7} + \frac{4}{7} - P(A \cap B) \]
This simplifies to:
\[ \frac{3}{7} = \frac{7}{7} - P(A \cap B) \]

### Step 6: Solve for \( P(A \cap B) \)
Rearranging gives:
\[ P(A \cap B) = \frac{7}{7} - \frac{3}{7} = \frac{4}{7} \]

### Step 7: Calculate \( P(A | B) \) and \( P(A' | B) \)
Using the formula for conditional probability:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{4}{7}}{\frac{4}{7}} = 1 \]
\[ P(A' | B) = \frac{P(A' \cap B)}{P(B)} = \frac{\frac{3}{7}}{\frac{4}{7}} = \frac{3}{4} \]

### Step 8: Calculate \( \frac{P(A | B)}{P(A' | B)} \)
Now we can find:
\[ \frac{P(A | B)}{P(A' | B)} = \frac{1}{\frac{3}{4}} = \frac{4}{3} \]

### Final Answer
Thus, the final answer is:
\[ \frac{P(A | B)}{P(A' | B)} = \frac{4}{3} \]

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