Question

If A and B are two events such that \(P(A)=\frac{1}{2}\), \(P(B)=\frac{1}{3}\) and \(P(A \cap B)=\frac{1}{4}\), then \(P\left(\frac{A'}{B}\right)=\)

• A. \(\frac{3}{4}\)
• B. \(\frac{1}{4}\)
• C. \(\frac{1}{2}\)
• D. \(\frac{5}{8}\)

Hint:

Use \(P(A'|B)=\frac{P(B)-P(A\cap B)}{P(B)}\).

Answer 1

The Correct Option is B

Step 1: Understand what is asked:
We need the probability that A does not happen, given that B has happened. This is a conditional probability, written \(P(A'|B)\).
A' means the complement of A, that is, the event that A does not occur.

Step 2: Recall the formula:
For a conditional probability we use \[ P(A'|B)=\frac{P(A' \cap B)}{P(B)} \]
The event B can be split into two parts that do not overlap: the part inside A and the part outside A. So \[ P(B)=P(A \cap B)+P(A' \cap B) \]
This gives \(P(A' \cap B)=P(B)-P(A \cap B)\).

Step 3: Find the numerator:
Put in the given values: \[ P(A' \cap B)=\frac{1}{3}-\frac{1}{4}=\frac{4-3}{12}=\frac{1}{12} \]

Step 4: Divide by P(B):
\[ P(A'|B)=\frac{1/12}{1/3}=\frac{1}{12}\times 3=\frac{1}{4} \]

Step 5: Check the options:
Option 1 (\(\frac{3}{4}\)) is actually \(P(A|B)=\frac{1/4}{1/3}\), which is the complement of our answer, so it is wrong.
Option 3 (\(\frac{1}{2}\)) is just \(P(A)\), and option 4 (\(\frac{5}{8}\)) does not come from any correct step. Only option 2 matches.

Final Answer:
\(P(A'|B)=\frac{1}{4}\), which is option 2. \[ \boxed{\frac{1}{4}} \]