In the world of probability, compound events are when two or more things are happening at once. We usually want to know the probability of all those things actually occurring, not each one of them at a time. For example, what is the probability that you forgot to do your homework and there will be a pop quiz in class?
We'll go over three different ways to compute these probabilities (organized lists, tree diagrams, and area models) and show you examples using each method.
Measuring Compound Events Using Organized Lists
Using the organized list method, you would list all the different possible outcomes that could occur. This can be difficult because there's a high probability that we will forget one or two options.
For example, if you flip a coin and roll a die, what is the probability of getting tails and an even number?
First, we need to start by listing all the possible outcomes we could get. (H1 means flipping heads and rolling a 1.)
| H1 | T1 |
| H2 | T2 |
| H3 | T3 |
| H4 | T4 |
| H5 | T5 |
| H6 | T6 |
There are twelve possible outcomes, and three of these outcomes give a desired outcome (tails plus an even number). These are T2, T4, and T6. So the probability is:
Example 1
In Clarajean's closet are four pairs of pants (black, white, grey, and brown), and five different shirts (blue, white, red, yellow, and purple). How many different outfits can she make with these options?
And, if she was to blindly pick a pair of pants and a shirt, what is the probability that the pants and shirt will be the same color?
First we need to list all the different outfits she can make, like white pants with a red shirt. The first color will be for the pants, and the second will be for the shirts.
| Black-Blue | Black-White | Black-Red | Black-Yellow | Black-Purple |
| White-Blue | White-White | White-Red | White-Yellow | White-Purple |
| Grey-Blue | Grey-White | Grey-Red | Grey-Yellow | Grey-Purple |
| Brown-Blue | Brown-White | Brown-Red | Brown-Yellow | Brown-Purple |
Whew, that was long and tedious, but it got the job done. There are 20 different outfits.
Only one of the outfits has matching pants and shirt (white-white). So, the probability of blindly picking pants and shirts of the same color is:
Example 2
If you toss a coin three times, what is the probability of flipping at least 2 heads?
Now, we are working with three different events (each flip counts as an individual event). To help keep this organized, we can list these in three columns.
| Flip 1 | Flip 2 | Flip 3 |
|---|---|---|
| H | H | H |
| H | H | T |
| H | T | H |
| H | T | T |
| T | H | H |
| T | H | T |
| T | T | H |
| T | T | T |
There are 8 different outcomes. These are the favorable ones: HHT, HTH, THH, and HHH (at least 2 heads includes flipping three). So the probability is:
Measuring Compound Events Using Tree Diagrams
Tree diagrams will give you the same answer as lists
Let's look at the coin and die example again: if you flip a coin and roll a die, what is the probability of getting tails and an even number?
We can chart all the possible outcomes by making a tree. The first set of "branches" will be all the possible outcomes of the first event. (It doesn't matter which event we put first, the total outcomes will be the same.) From each of those outcomes, draw branches for all the possibilities of the second event.
It could look like this:
Or this:
By counting the smallest branches, we see that there are 12 possibilities. So the probability of flipping a tail and rolling an even number is:
Example 1
In Clarajean's closet are four pairs of pants (black, white, grey, and brown), and five different shirts (blue, white, red, yellow, and purple). How many different outfits can she make with these options?
And, if she was to blindly pick a pair of pants and a shirt, what is the probability that the pants and shirt will be the same color?
If we start with the shirts, then add the pants, our tree diagram should look something like this:
Only one branch has matching pants and shirt (white pants and white shirt). So, the probability of blindly picking pants and shirts of the same color is:
Example 2
If you toss a coin three times, what is the probability of flipping at least 2 heads?
With three events, we will have three sets of branches on our tree.
There are eight different possibilities, four of which give at least two heads. So the probability is:
Measuring Compound Events Using Area Models
The final way we will chart these problems is the area model.We're smitten with this method because it's hard to forget an outcome using it.
Example: if you flip a coin and roll a die, what is the probability of getting tails and an even number?
Start by making a table with the outcomes of one event listed on the top and the outcomes of the second event listed on the side. Fill in the cells of the table with the corresponding outcomes for each event. We can shade the cells that fit our probability.
There are twelve cells, of which three are shaded. So the probability is:
Example 1
Going back to Clarajean's closet, she has four pairs of pants (black, white, grey, and brown), and five different shirts (blue, white, red, yellow, and purple). How many different outfits can she make with these options?
And, again, if she was to blindly pick a pair of pants and a shirt, what is the probability that the pants and shirt will be the same color?
Here's our table. We shaded the favorable outcome.
There are 20 different cells, which means 20 different options (4x5=20). One cell is shaded to represent the one time when she could pick the same color for both shirt and pants. Based on the calculation below, the probability that she'll pick the same color for both is 5%.
Example 2
If you toss a coin three times, what is the probability of flipping at least 2 heads?
Three events make the area model a bit more complicated since there are only two places to list the events (top and side). To work around this, draw one table for the first two events,
Now use those outcomes (HH, HT, TH, TT) for the rows or columns of a second table.
In the final table, there are eight different cells, four of which are shaded.