Lesson 68: Mutually Exclusive and Inclusive Events

New Concept

Two events that cannot both occur in the same trial or experiment are mutually exclusive events, or disjoint events.

What’s next

Next, you'll apply these rules to calculate probabilities in various scenarios, from rolling dice to analyzing survey data.

Property

If $A$ and $B$ are mutually exclusive events, then

Explanation

Think about flipping a single coin—you can get heads, or you can get tails, but you can't possibly get both at the very same time. These types of events are 'mutually exclusive' because they can never happen together. Calculating the probability of one OR the other happening is super easy: you just add their individual probabilities together!

Examples

• What is the probability of rolling a sum of 6 or a sum of 11 with two dice? $\operatorname{P}(6 \text{ or } 11) = \operatorname{P}(6) + \operatorname{P}(11) = \frac{5}{36} + \frac{2}{36} = \frac{7}{36}$.
• From a deck of cards, what's the chance of drawing a 7 or a Jack? $\operatorname{P}(7 \text{ or Jack}) = \operatorname{P}(7) + \operatorname{P}(\text{Jack}) = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \frac{2}{13}$.
• A weather forecast gives a 30% chance of only rain and a 25% chance of only snow. What is the chance of rain or snow? $\operatorname{P}(\text{rain or snow}) = 0.30 + 0.25 = 0.55$.

Let's see what happens when two outcomes can't possibly happen at the same time. This example uses the first key idea from our lesson, mutually exclusive events, where we can simply add probabilities together.

Example Problem

What is the probability of rolling either a sum of 5 or a sum of 10 using two different number cubes?

Step-by-Step

1. First, we need to recognize that it is impossible to roll both a sum of 5 and a sum of 10 in a single roll. This means the events are mutually exclusive.
2. We can create a table of all 36 possible outcomes to find the probabilities for each event.

Property

If $A$ and $B$ are inclusive events, then

Explanation

Imagine drawing a card from a deck. Could it be a queen and also a spade? Absolutely! The Queen of Spades fits both. These are 'inclusive' events because they can overlap. To find the total probability without unfairly double-counting the overlap, we add the chances of each event and then subtract the probability of them happening together.

Examples

• What is the probability of drawing a Heart or a King from a deck? $\operatorname{P}(\text{H or K}) = \operatorname{P}(\text{H}) + \operatorname{P}(\text{K}) - \operatorname{P}(\text{H and K}) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$.
• With two dice, what is the probability of rolling at least one odd number or a sum of 8? $\operatorname{P}(\text{odd or 8}) = \frac{27}{36} + \frac{5}{36} - \frac{2}{36} = \frac{30}{36} = \frac{5}{6}$.
• An MP3 player has pop and country artists, including 7 crossover artists. What is the chance of playing a pop or country song? $\operatorname{P}(\text{pop or country}) = \frac{25}{80} + \frac{26}{80} - \frac{7}{80} = \frac{44}{80} = \frac{11}{20}$.

Now, what if two events can happen at the same time? This example covers our second key idea, inclusive events, where we must avoid double-counting.

Example Problem

What is the probability of rolling doubles or a sum of 6 using two number cubes?

Step-by-Step

1. First, determine if the events are inclusive. We need to list the outcomes for each event.
2. Outcomes for rolling doubles: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6). So, $\operatorname{P}(\text{doubles}) = \frac{6}{36}$.
3. Outcomes for rolling a sum of 6: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1). So, $\operatorname{P}(\text{sum of } 6) = \frac{5}{36}$.
4. Check for overlap. The outcome (3, 3) appears in both lists. This means the events are inclusive, and the probability of the overlap is $\operatorname{P}(\text{doubles and sum of } 6) = \frac{1}{36}$.
5. Use the formula for inclusive events, which subtracts the overlap to prevent counting it twice.