Let A and B be two events such that P(A|B) = P(A'|B') = p and P(B) = 0.05. The value of p so that P(B|A) =0.9 is

Options

A. `21//430`
B. `173//430`
C. `171//430`
D. `95//430`

Answer

To solve the problem step by step, we will use the given information about the probabilities of events A and B.

### Step 1: Understand the Given Information
We know:
- \( P(A|B) = p \)
- \( P(A'|B') = p \)
- \( P(B) = 0.05 \)
- We want to find \( p \) such that \( P(B|A) = 0.9 \).

### Step 2: Use the Definition of Conditional Probability
From the definition of conditional probability, we can express \( P(A|B) \) and \( P(B|A) \) as follows:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
\[ P(B|A) = \frac{P(B \cap A)}{P(A)} \]

Since \( P(A \cap B) = P(B \cap A) \), we can denote \( P(A \cap B) = X \).

### Step 3: Express \( P(A|B) \) in Terms of \( X \)
Substituting the known values into the equation for \( P(A|B) \):

\[ p = \frac{X}{0.05} \quad \text{(1)} \]

### Step 4: Express \( P(B|A) \) in Terms of \( P(A) \)
Let \( P(A) = Y \). Then, we can express \( P(B|A) \):

\[ 0.9 = \frac{X}{Y} \quad \text{(2)} \]

### Step 5: Use the Complementary Probability
Using the information about \( P(A'|B') \):

\[ P(A'|B') = \frac{P(A' \cap B')}{P(B')} \]

Using De Morgan's theorem, we can express \( P(A' \cap B') \):

\[ P(A' \cap B') = P((A \cup B)') = 1 - P(A \cup B) \]

Using the formula for the union of two events:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) = Y + 0.05 - X \]

Thus,

\[ P(A' \cap B') = 1 - (Y + 0.05 - X) = 0.95 - Y + X \]

Now substituting into the equation for \( P(A'|B') \):

\[ p = \frac{0.95 - Y + X}{0.95} \quad \text{(3)} \]

### Step 6: Substitute \( X \) and \( Y \) in Terms of \( p \)
From equation (1), we have:

\[ X = 0.05p \]

From equation (2):

\[ Y = \frac{X}{0.9} = \frac{0.05p}{0.9} = \frac{p}{18} \]

### Step 7: Substitute \( X \) and \( Y \) into Equation (3)
Substituting \( X \) and \( Y \) into equation (3):

\[ p = \frac{0.95 - \frac{p}{18} + 0.05p}{0.95} \]

### Step 8: Solve for \( p \)
Multiplying both sides by \( 0.95 \):

\[ 0.95p = 0.95 - \frac{p}{18} + 0.05p \]

Combining like terms:

\[ 0.95p + \frac{p}{18} - 0.05p = 0.95 \]

This simplifies to:

\[ 0.9p + \frac{p}{18} = 0.95 \]

Finding a common denominator (which is 18):

\[ \frac{16.2p + p}{18} = 0.95 \]

\[ \frac{17.2p}{18} = 0.95 \]

Now, multiply both sides by 18:

\[ 17.2p = 17.1 \]

Finally, solving for \( p \):

\[ p = \frac{17.1}{17.2} = \frac{171}{172} \]

### Final Answer
Thus, the value of \( p \) is \( \frac{171}{172} \).

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