Common Core Standards: Math
9. Use permutations and combinations to compute probabilities of compound events and solve problems.
You have the chance to participate in a blind taste test of some pizza with different toppings: pickle (P), blueberry (B), gummy bear (G), and Oreo crumbs (O). You can only taste two slices, and the order in which you taste the pizza matters for the study. The possible outcomes for the study are:
There are 12 distinct outcomes, and the probability of observing any one outcome is 1 in 12.
Let's say you have a really lazy sampler, and he says the order doesn't matter. This time, the possible outcomes are:
There are only 6 distinct outcomes now. What gives?
In the first sample, where order doesn't matter, permuting the different events leads to a greater set of outcomes. We say you can taste 1 of 12 possible permutations of pizza. When order doesn't matter, you can taste 1 of 6 possible combinations of pizza.
Recognizing this difference, students should learn the formulas to calculate the numbers of possible outcomes in each of these cases. Writer's cramp will probably set in by the time they get to 100 different outcomes. We should start with the simpler formula first, the formula for the number of permutations.
We say they are n things that can fill r slots. We write this as nPr, and there's a handy formula to calculate it. (Be sure to introduce factorials if they don't understand them. Otherwise, they might think you're just really excited about numbers. That might be true, but it's irrelevant.)
In our case, we have n = 4 things (P, B, G, and O) to fill r = 2 slots.
There are 12 possible outcomes, all of which we listed.
For combinations, in which the order does not matter, the total number of outcomes nCr for an experiment with n events and r slots is given by:
In our case, there are n = 4 possible events and can fill r = 2 slots.
We listed 6 possible outcomes.
Finally, students should be able to connect the number of possible outcomes back to probability. In particular, the students should be able to recognize when to apply the permutation or combination formula to answer the problem at hand. Probably significantly less gross than a pickle pizza.