Question

Given \(P(A) = p_1, P(B) = p_2\) and \(P(AB) = p_3\). Then \(P(A' \cap B')\) will be:

• A. \(1 - p_1 - p_2\)
• B. \(1 - p_3\)
• C. \(1 - p_1 p_2 p_3\)
• D. \(1 - p_1 - p_2 + p_3\)

Hint:

To quickly find the probability that neither event occurs, calculate \(1 - P(\text{at least one event occurs})\).
This is always \(1 - P(A \cup B)\), which expands easily using the addition rule.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem belongs to probability theory, specifically dealing with the properties of set operations, De Morgan's laws, and the addition theorem of probability.
The objective is to express the probability of the simultaneous non-occurrence of two events, \(A\) and \(B\), in terms of their individual and joint probabilities.

Step 2: Key Formula or Approach:
According to De Morgan's Laws in set theory, the intersection of the complements of two sets is equal to the complement of their union:
\[ A' \cap B' = (A \cup B)' \]
Taking the probability on both sides, we get:
\[ P(A' \cap B') = P((A \cup B)') = 1 - P(A \cup B) \]
The Addition Theorem of Probability states that:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Step 3: Detailed Explanation:
We are given the following individual probabilities:
\[ P(A) = p_1 \]
\[ P(B) = p_2 \]
The probability of their joint occurrence (intersection) is given by:
\[ P(AB) = P(A \cap B) = p_3 \]
First, we calculate the probability of the union of the two events using the addition theorem:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the given values into this expression yields:
\[ P(A \cup B) = p_1 + p_2 - p_3 \]
Next, we determine the probability of the event that neither \(A\) nor \(B\) occurs.
Using the complement rule, we subtract the probability of the union from 1:
\[ P(A' \cap B') = 1 - P(A \cup B) \]
\[ P(A' \cap B') = 1 - (p_1 + p_2 - p_3) \]
Expanding the brackets, we obtain:
\[ P(A' \cap B') = 1 - p_1 - p_2 + p_3 \]
This matches the expression in the fourth option.

Step 4: Final Answer:
Therefore, the correct option is (D).