Question

Let \( A \) and \( B \) be two events. If \( P(A|B)=0.4 \), \( P(A|B')=0.7 \) and \( P(B)=0.7 \), then \( P(A) \) is

• A. \(0.44 \)
• B. \(0.54 \)
• C. \(0.49 \)
• D. \(0.5 \)
• E. \(0.65 \)

Hint:

Always split probability using \(B\) and its complement.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem requires the use of the Law of Total Probability. This law states that the probability of an event A can be found by summing the probabilities of A occurring along with each event in a partition of the sample space. Here, the events B and B' (the complement of B) form a partition.
Step 2: Key Formula or Approach:
The Law of Total Probability states: \[ P(A) = P(A \cap B) + P(A \cap B') \] Using the definition of conditional probability, \(P(X \cap Y) = P(X|Y)P(Y)\), we can write this as: \[ P(A) = P(A|B)P(B) + P(A|B')P(B') \] We also need the formula for the complement: \(P(B') = 1 - P(B)\).
Step 3: Detailed Explanation:
We are given:
\(P(A|B) = 0.4\)
\(P(A|B') = 0.7\)
\(P(B) = 0.7\)
First, we need to find \(P(B')\): \[ P(B') = 1 - P(B) = 1 - 0.7 = 0.3 \] Now, we can apply the Law of Total Probability: \[ P(A) = P(A|B)P(B) + P(A|B')P(B') \] Substitute the known values into the formula: \[ P(A) = (0.4)(0.7) + (0.7)(0.3) \] \[ P(A) = 0.28 + 0.21 \] \[ P(A) = 0.49 \] Step 4: Final Answer:
The value of P(A) is 0.49.