In statistics and probability, the intersection sign is written as \(\cap\). For events \(A\) and \(B\), the notation \(A \cap B\) means the set of outcomes that belong to both events—equivalently, “\(A\) and \(B\) happen together.”
Meaning of the intersection sign (∩)
Definition: \(A \cap B\) is the event that occurs when both \(A\) occurs and \(B\) occurs. It contains only the outcomes common to \(A\) and \(B\).
| Symbol | Name | Meaning (events as sets of outcomes) |
|---|---|---|
| \(\cap\) | Intersection | \(A \cap B\): outcomes in both \(A\) and \(B\) (both occur) |
| \(\cup\) | Union | \(A \cup B\): outcomes in \(A\) or \(B\) (at least one occurs) |
| \(A^c\) | Complement | \(A^c\): outcomes not in \(A\) (event \(A\) does not occur) |
| \(A \setminus B\) | Difference | \(A \setminus B\): outcomes in \(A\) but not in \(B\) |
How to compute \(P(A \cap B)\) using conditional probability
Conditional probability is defined (when \(P(B) > 0\)) by
\[ P(A \mid B)=\frac{P(A \cap B)}{P(B)}. \]
Solving this definition for the intersection probability gives the multiplication rule:
\[ P(A \cap B)=P(A \mid B)\cdot P(B). \]
Independent events case: If \(A\) and \(B\) are independent, then \(P(A \mid B)=P(A)\), so \[ P(A \cap B)=P(A)\cdot P(B). \]
Worked example with the intersection sign
Suppose two events in a statistics experiment satisfy: \(P(B)=0.40\) and \(P(A \mid B)=0.70\). Find \(P(A \cap B)\).
- Use the multiplication rule for the intersection sign: \[ P(A \cap B)=P(A \mid B)\cdot P(B). \]
- Substitute the given values: \[ P(A \cap B)=0.70\cdot 0.40. \]
- Multiply: \[ P(A \cap B)=0.28. \]
Therefore, the probability that both events occur (the intersection \(A \cap B\)) is \(0.28\).
Visualization: Venn diagram showing the intersection (∩)
Common follow-up: relating intersection to union
Once \(P(A \cap B)\) is known, the probability of the union (at least one event occurs) can be computed by
\[ P(A \cup B)=P(A)+P(B)-P(A \cap B). \]
Important condition: \(P(A \mid B)\) is defined only when \(P(B) > 0\); otherwise the ratio \(\frac{P(A \cap B)}{P(B)}\) is not defined.