Analyzing Probability Statements for Events A and B

Let's break down each probability statement regarding two events A and B to determine their correctness.

Statement 1: P(A occurs but not B) = P(A) – P(B) if B ⊂ A

The event "A occurs but not B" is represented as \(A \cap B'\). The general formula for the probability of \(A \cap B'\) is \(P(A \cap B') = P(A) - P(A \cap B)\).

The statement includes a condition: if \(B \sub A\). This means that every outcome in event B is also an outcome in event A. In this case, the intersection of A and B, \(A \cap B\), is simply event B itself.

So, if \(B \sub A\), then \(A \cap B = B\).

Substituting this into the general formula, we get:

\(P(A \cap B') = P(A) - P(B)\)

This matches the formula given in statement 1. Therefore, statement 1 is correct under the given condition.

Statement 2: P(A alone or B alone occurs) = P(A) + P(B) – P(A ∩ B)

The phrase "A alone occurs" refers to the event \(A \cap B'\) (A occurs but not B). The phrase "B alone occurs" refers to the event \(B \cap A'\) (B occurs but not A).

The event "A alone or B alone occurs" is the union of these two mutually exclusive events: \((A \cap B') \cup (B \cap A')\).

Since these two events \((A \cap B')\) and \((B \cap A')\) are mutually exclusive (they cannot happen at the same time), the probability of their union is the sum of their individual probabilities:

\(P((A \cap B') \cup (B \cap A')) = P(A \cap B') + P(B \cap A')\)

Using the formulas for \(P(A \cap B')\) and \(P(B \cap A')\):

• \(P(A \cap B') = P(A) - P(A \cap B)\)
• \(P(B \cap A') = P(B) - P(A \cap B)\)

Adding these together:

\(P(\text{A alone or B alone}) = (P(A) - P(A \cap B)) + (P(B) - P(A \cap B))\)

\(P(\text{A alone or B alone}) = P(A) + P(B) - 2 \cdot P(A \cap B)\)

The formula given in statement 2 is \(P(A) + P(B) – P(A ∩ B)\), which is actually the formula for \(P(A \cup B)\). This is different from \(P(A) + P(B) - 2 \cdot P(A \cap B)\).

Therefore, statement 2 is incorrect.

Statement 3: P(A ∪ B) = P(A) + P(B) if A and B are mutually exclusive

The general addition rule for any two events A and B is:

\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

The statement includes a condition: if A and B are mutually exclusive. Mutually exclusive events are events that cannot occur at the same time, meaning their intersection is an empty set (\(A \cap B = \emptyset\)).

For mutually exclusive events, the probability of their intersection is 0:

\(P(A \cap B) = 0\)

Substituting \(P(A \cap B) = 0\) into the general addition rule:

\(P(A \cup B) = P(A) + P(B) - 0\)

\(P(A \cup B) = P(A) + P(B)\)

This matches the formula given in statement 3. Therefore, statement 3 is correct under the condition that A and B are mutually exclusive.

Summary of Correct Statements

Based on the analysis:

• Statement 1 is correct.
• Statement 2 is incorrect.
• Statement 3 is correct.

Thus, the correct statements are 1 and 3 only.

Revision Table: Probability Rules

Additional Information: Set Notation in Probability

Understanding the set notation used in probability helps clarify the events being described:

• \(A \cap B\): Represents the event that both A and B occur. This is the intersection of A and B.
• \(A \cup B\): Represents the event that A occurs, or B occurs, or both occur. This is the union of A and B.
• \(A'\) or \(B'\): Represents the complement of A (or B), meaning the event that A (or B) does not occur.
• \(A \cap B'\): Represents the event that A occurs but B does not occur.
• \(B \cap A'\): Represents the event that B occurs but A does not occur.
• \(B \sub A\): Means event B is a subset of event A, implying that whenever B occurs, A must also occur.

These notations are crucial for correctly interpreting and applying probability rules.