Consider a chance experiment where you roll the die below and spin this spinner.
Each of these spinning sections is not equally likely. But there are, in fact, four sections. How many different outcomes of this experiment are possible?
One way to visualize this is with something we call a tree diagram. To start, we're going to enumerate all the possible outcomes that could happen from the first chance experiment, which is the rolling the die. So, we're going to make a tree with six possibilities for the die: 1, 2, 3, 4, 5, and 6.
Next in the tree are all the possibilities for the spinner, for each possibility for the die. For instance, if a 1 is rolled, the spinner might spin A, B, C, or D. If a 2 is rolled, the spinner might land on A, B, C, or D, etc. Each outcome represents a different path: one path is 1, C, for example. Another path is 1, D, and so forth. What you end up seeing are 24 different outcomes. There are six branches for the die, each of which has four outcomes for the spinner. So it's like calculating 6 times 4.
In fact, that's what the fundamental counting principle says. It says if you do two chance experiments, A and B, then experiment A has x potential outcomes and experiment B has y potential outcomes. It follows, then, that there are x times y potential outcomes when experiments A and B are performed together.
The fundamental counting principal can actually extend beyond just two experiments. It can extend to three, four, five, or however many experiments are being conducted together, by simply multiplying the number of potential outcomes for each consecutive experiment. In the example above, there were 6 outcomes for the die and 4 outcomes for the spinner, so there were 6 times 4, or 24 potential outcomes.
IN CONTEXT
A family is going to have three children. How many different orderings of children are there in terms of boys and girls?
The first child could be a boy or a girl. If you start with a boy, you could have another boy, and the third child could be a girl. Another outcome is that you could have three boys.
If you start with a girl, the second and third child could both be boys. Another outcome could be that you have two girls, then have a boy.
As you can see, there are many different outcomes. To find all possible outcomes, you can create a tree diagram.
Looking at all the tree diagram branches here, you can see there are eight possible outcomes.
An easier way to do it would be to use the Fundamental Counting Principle. For the first child, there are two options; for the second child, two options; and for the third child, two options.
2 times 2 times 2 equals 8 outcomes. The tree diagram isn't really necessary. The number of choices for each of the children multiplied by each outcome for the children is what is needed.
Source: Adapted from Sophia tutorial by Jonathan Osters.