Question

The probability that both S and T occur, the probability that S occurs and T does not, and the probability that T occurs and S does not are all equal to p. The probability that either S or T occurs is

• A. p
• B. 2p
• C. 3p
• D. \(3p^2\)

Hint:

A Venn diagram partitions the union of two sets into exactly three mutually exclusive parts:
- Only S
- Only T
- Both S and T
Since each part has a probability of \(p\), the sum of the entire union is simply \(p + p + p = 3p\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We can solve this problem using a Venn diagram.
An event "either S or T occurs" is represented by the union of the two sets, \(S \cup T\).
This union can be decomposed into three mutually exclusive, disjoint regions.

Step 2: Detailed Explanation:
Let us define the three disjoint regions that make up the union \(S \cup T\):
1. The region where \(S\) occurs and \(T\) does not: \(S \cap T^c\)
2. The region where \(T\) occurs and \(S\) does not: \(T \cap S^c\)
3. The region where both \(S\) and \(T\) occur: \(S \cap T\)
Since these three regions do not overlap, we can express the probability of their union as the sum of their individual probabilities:
\[ P(S \cup T) = P(S \cap T^c) + P(T \cap S^c) + P(S \cap T) \]
According to the problem statement, the probabilities of all three events are equal to \(p\):
- \(P(S \cap T) = p\)
- \(P(S \cap T^c) = p\)
- \(P(T \cap S^c) = p\)
Now, substitute these values into the union equation:
\[ P(S \cup T) = p + p + p = 3p \]
Thus, the probability that either \(S\) or \(T\) occurs is \(3p\).

Step 3: Final Answer:
The correct option is (C).