This tutorial will discuss fundamental counting principle by focusing on:
Consider a chance experiment where the die below is rolled, and this spinner is spun.
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. All possible outcomes are first enumerated. All the outcomes that could happen from the first chance experiment, which is the rolling the die.
A tree with six possibilities will be noted, since all possibilities for the die 1, 2, 3, 4, 5, and 6.
Then if a 1 is rolled, the spinner might spin A, B, C, or D. If a 2 is rolled, the spinning might land on A, B C, or D, etc.
First in the tree are the possibilities for the die, then the possibilities for the spinner off of each possibility for the die. When the total number of paths are counted, starting from back forward, this is one path.
One path is 1, C. And then 1 and D is another path. What you end up seeing is 24 different outcomes.
Twenty-four sounds like a reasonable number when you think about a die and this spinner. There are six branches for the die, each of which has four outcomes for the spinner. So it's like doing 6 times 4.
In fact, that's what the fundamental counting principle says. It says if you do two chance experiments, A and B, the experiment A has x potential outcomes. And experiment b has y potential outcomes. Then there are x times y potential outcomes when A and B are done together.
The fundamental counting principal can actually extend this beyond just two experiments. It can extend it to three, or four, or five, or however many experiments being conducted together by simply multiplying the number of potential outcomes for each consecutive experiment. In the case of the die, there were 6 outcomes and 4 outcomes for the spinner, therefore there were 24 potential outcomes.
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 a boy and a girl again. If you start with a girl you could have a boy and girl again. And then the third child if you start boy, boy could be another boy or it could be a girl. If you start boy, girl it could be a boy or a girl,etc.
Looking at all the tree diagram branches here, you can see there are 1, 2, 3, 4, 5, 6, 7, 8 if you counted on the tree diagram
An easier way to do it would have been to realize first child two options, second child two options, third child two options. 2 times 2 times 2 is 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.
The fundamental counting principle is used to determine the total number of different outcomes that could result from several chance experiments done either at the same time or one right after the other.
The number of potential outcomes is equal to the product of the number of trials for each experiment. So we did 6 times 4. We did 3 times 3 times 3. And you know what, tree diagrams really are useful tools for visualizing all the different permutations, all the different paths, that these chance experiments could take.
So we talked about the fundamental counting principle and the visualization based on tree diagrams.
Source: This work is adapted from Sophia author jonathan osters.
A way to visualize the different "paths" that a sequence of chance experiments could take.
If chance experiment A has m possible outcomes for one trial, and chance experiment B has n possible outcomes for its trial (independent of the first trial), then there are mxn potential outcomes when A and B are done together.