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Compound events involve two (or more) independent events happening together. In these events, the compound probability is computed by multiplying the probabilities of each individual event.

**Example 1**

Consider the event of flipping a coin, rolling a die, and drawing a card from a deck of cards. Find the probability of flipping heads, rolling an even number, or drawing a 7.

To compute this probability, it may help to think of the event as being three separate events: 1) flipping heads, 2) rolling an even number, and 3) drawing a 7. In order to compute the compound probability, or the probability that you complete all three of these events together, can be computed by finding each event's probability and then multiplying them.

**Event 1**

P(flipping heads) = 1/2

because there are two possible outcomes and one of them is heads

**Event 2**

P(rolling an even number) = 3/6 = 1/2

because there are 6 possible outcomes on a die and 3 of them are even

**Event 3**

P(drawing a 7) = 4/52

because there are 52 cards in a standard deck of playing cards and 4 of them are 7's - one for each suit

Now we can take each of these probabilities and multiply them to find the compound probability:

1/2 * 1/2 * 4/52 = 1/52 or approximately .019

**Example 2**

You're picking out ice cream at a shop that offers 3 different flavors of ice cream (vanilla, chocolate, strawberry) and 5 toppings (sprinkles, chocolate syrup, cherries, chocolate chips, whip cream). If you randomly choose an ice cream flavor and one topping, what is the probability that you pick vanilla ice cream without chocolate?

Again, we want to imagine this as two independent events: 1) picking vanilla ice cream and 2) picking a topping that isn't chocolate. Note that there are two (2) chocolate options in the list (chocolate syrup and chocolate chips). So we begin by finding the probability for each event.

**Event 1**

P(vanilla) = 1/3

because there are 3 flavors to choose from and 1 is vanilla

**Event 2**

P(not chocolate) = 3/5

because there are 5 toppings to choose from and 3 of them are not chocolate

Then we use the multiplication rule to compute the compound probability of these two events happening together:

1/3 * 3/5 = 3/15 = 1/5 or .20

- Last Updated: Apr 19, 2024 3:09 PM
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