Non-Mutually Exclusive Events A and B

What Does Non-Mutually Exclusive Mean?

We learned about mutually exclusive events that can't happen together (like turning left and right at the same time). Now, let's talk about Non-Mutually Exclusive Events. These are two (or more) events that CAN happen at the same time in a single experiment.

This means it's possible to get an outcome that belongs to event A and also belongs to event B.

Simple Examples:

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

   • Event A: Getting a Heart ($\heartsuit$).
   • Event B: Getting a King. Can events A and B happen together? Absolutely! There's a card that is both a Heart and a King: the King of Hearts ($K\heartsuit$). Since they can happen together, events A and B are non-mutually exclusive.

2. Rolling a Die (once):

   • Event A: Getting an even number ($\{2, 4, 6\}$).
   • Event B: Getting a number greater than 3 ($\{4, 5, 6\}$). Can these happen together? Yes! The numbers $4$ and $6$ are both even and greater than 3. So, events A and B are non-mutually exclusive.

The Intersection is Important!

In non-mutually exclusive events, there's a part that belongs to both events simultaneously. This part is called the intersection.

Because there is an intersection, the probability of event A AND B happening together is greater than zero.

Or using the intersection symbol:

This is very different from mutually exclusive events, where $P(A \cap B) = 0$.

Calculating P(A OR B) for Non-Mutually Exclusive Events

Since there's a chance that events A and B can happen together, we can't just add $P(A) + P(B)$ to find $P(A \text{ or } B)$.

Why not? Because if we simply add them, the intersection part ($A \cap B$) gets counted twice, once in $P(A)$ and again in $P(B)$.

To get the correct calculation, we must subtract the probability of the intersection that was double-counted. This gives us the General Addition Rule for probability:

Or using the union and intersection symbols:

This formula works generally, for both mutually exclusive and non-mutually exclusive events. (If they are mutually exclusive, $P(A \cap B)$ is zero, so the formula simplifies back to $P(A \cup B) = P(A) + P(B)$).

Calculation Example

Let's use the card example:

• Event A: Getting a Heart ($\heartsuit$). There are 13 Hearts in 52 cards. $P(A) = 13/52$.
• Event B: Getting a King. There are 4 Kings in 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.