Overlapping Events A and B

Two Events That Share an Outcome

Two events are not mutually exclusive when they can occur in the same trial. In set language, at least one outcome belongs to both events.

Their outcome sets therefore have a nonempty intersection.

Two examples:

1. Drawing a Card: You draw one card from a standard deck.

   • Event $A$: Getting a Heart ($\heartsuit$).
   • Event $B$: Getting a King. The King of Hearts ($K\heartsuit$) belongs to both events. Therefore, events $A$ and $B$ are not mutually exclusive.

2. Rolling a Die (once):

   • Event $A$: Getting an even number ($\{2, 4, 6\}$).
   • Event $B$: Getting a number greater than $3$ ($\{4, 5, 6\}$). The outcomes $4$ and $6$ belong to both events. Therefore, events $A$ and $B$ are not mutually exclusive.

Intersection of Overlapping Events

For non-mutually exclusive events, some outcomes may belong to both $A$ and $B$. These shared outcomes form the intersection $A \cap B$.

The definition is about the outcome sets themselves: the two events are not mutually exclusive when their intersection is nonempty.

In a finite sample space where every elementary outcome has positive probability, a nonempty intersection also has positive probability:

For mutually exclusive events, the intersection has probability $P(A \cap B) = 0$.

Calculating the Probability of a Union

Because $A$ and $B$ may occur together, adding $P(A) + P(B)$ would count the overlap twice.

That sum would count the intersection $A \cap B$ twice, once in $P(A)$ and once in $P(B)$.

We therefore subtract the probability of the intersection once. This gives the general addition rule:

Using union and intersection symbols:

This formula applies to both cases. For mutually exclusive events, $P(A \cap B)=0$, so it simplifies to $P(A \cup B) = P(A) + P(B)$.

Probability of Drawing a Heart or a King

Use the card example:

• Event $A$: Getting a Heart ($\heartsuit$). There are $13$ Hearts among the $52$ cards. $P(A) = 13/52$.
• Event $B$: Getting a King. There are $4$ Kings among the $52$ cards. $P(B) = 4/52$.
• Event $A$ and B: Getting the King of Hearts ($K\heartsuit$). There is only $1$ King of Hearts. $P(A \cap B) = 1/52$.

So, the probability of getting a Heart OR a King is:

See? We subtract $1/52$ so the King of Hearts isn't counted twice.

Exercise

A fair six-sided die is rolled once. Let $A=\{2,4,6\}$ be the event “the result is even” and $B=\{5,6\}$ the event “the result is greater than $4$.”

1. Explain why $A$ and $B$ are not mutually exclusive.
2. Find $P(A \cup B)$ using the general addition rule.
3. Verify the value of $P(A \cup B)$ by listing the outcomes in the union.

Worked Solution

The intersection contains one shared outcome, namely $6$:

The events are therefore not mutually exclusive. Their probabilities and the intersection probability are

Apply the general addition rule:

Directly listing the union gives $A \cup B=\{2,4,5,6\}$, which also contains $4$ of the $6$ possible outcomes. Both methods therefore give $2/3$.