Combinations

The page on proportions and probabilities teaches the basics of probability. Moving onwards.....

The Basic Concept of Probability

Your students might already have the concept of probability, or maybe you can just tell them the concept. However, to teach the concept of probability using my method, you have to give them a problem for which the concept of probability is the answer. This is one such problem. I have 3 greens and 1 red in my pocket. (They can be anything that is identical except for color.) I mix them around, then pick one. What can we say about what I will pick?

One answer is that it will be green. A second answer is that it could be green and it could be red. The second statement is true, but it does not say everything that can be said about which color would come out. The second statement mentions what we already know, but it isn't quite correct -- as the second statement indicates, the thing could be red. So what can we say about what will come out? We want some statement that captures both of these statements -- that it could be red, but green has some advantage over red.

The answer, whether or not the student can put words to it, is that green is more probable. Put another way, there is a higher chance of green, or green is more likely.

Teaching Multiplication

What is the probability of rolling a pair of sixes with two dice? If the probability of one six with one die is 1 to 5, then the probability of two sixes with two dice is 1 to 35. The point I am making is that there is no easy formula connecting the two, with this measure of probability.

There is an easy connection when probabilities are measured by proportions -- multiplication. The probability of one six with one die is 1/6 and the probability of two sixes is 1/6 * 1/6, or 1/36. Put technically, the probability of two independent events equals the product of the probabilities of the two events (when probability is measured as a proportion).

You could just tell students this and hope for the best. But then they are memorizing it, not understanding it. And you have the very small problem of the independent-events caveat, which makes perfect sense from the standpoint of understanding but poses a serious problem for memory.

Is there any way to teach understanding through problems? I suspect the best hope is the combination problem. Suppose Lilly has 3 hats and 3 coats. How many outfits (of hats and coats) can she wear? This is a counting problem, first of all. Second, it can be turned into a pattern problem -- what is the formula for the answer? And third, it is a probability problem -- what is the probability of any given combination of hat and coat, assuming she chooses randomly.

Warm-up

In teaching, I used this first problem as a warm-up. The goal was to list combinations, but it wasn't important that the students get them all
Lusac has 3 shoes that fit his left foot and 4 shoes that fit his right foot. Unfortunately, they are all different, so Lusac cannot wear a matching pair of shoes. But, given that his right shoe will look different from his left shoes, how many different combinations can Lusac wear?

Basic Problems

The next problems there were supposed to get all of the combinations.

It's gym day. Rebecca has two different pairs of sweat pants she could wear, and she has 3 different t-shirts she could wear. How many different outfits can she wear?

Rachel has two different pairs of sweat pants she could wear, and she has 4 different t-shirts she could wear. How many different outfits can she wear for gym day?

Peter has three different pairs of sweat pants he could wear, and he has 3 different t-shirts he could wear. How many different outfits can he wear for gym day?

Alyssa has 3 different pairs of sweat pants she could wear, and she has 6 different t-shirts she could wear. How many different outfits can she wear for gym day?

The General Pattern

Now the problem is to express the pattern abstractly. The abstract pattern is needed for the second problem. And really, the student should see the two problems as identical.

Ashlet has m different pairs of sweat pants she could wear, and she has n different t-shirts she could wear. How many different outfits can she wear for gym day?

Kaini has 27 different pairs of sweat pants she could wear, and she has 45 different t-shirts she could wear. How many different outfits can she wear for gym day?

Finally, if a student notices that the pattern is multiplication, that is very good. But that doesn't mean that the student understands why multiplication works. So then the question is, why does multiplication work? Explain.

On the Lighter Side

These are supposed to be a little humorous. But they also go with expressing the general pattern and can be used before that or with that as needed.

Leslie has 1 pair of sweat pants and 2 t-shirts. How many different outfits can she wear for gym day?
One of her t-shirts is destroyed. Now Leslie has 1 pair of sweat pants and 1 t-shirt. How many different outfits can she wear for gym day?

Another t-shirt is destroyed. Now Leslie has 1 pair of sweat pants and 0 t-shirts. How many different outfits can she wear for gym day?

Her grandmother gives Leslie 24 pairs of sweat pants. Now Leslie has 25 pairs of sweat pants and 0 t-shirts. How many different outfits can she wear for gym day?

Her mother goes to a yard sale and buys 47 pairs of sweat pants. Now Leslie has 72 pairs of sweat pants and 0 t-shirts. How many different outfits can she wear for gym day?

Her father buys a company that makes sweat pants. Now Leslie has 1079 pairs of sweat pants and 0 t-shirts. How many different outfits can she wear for gym day?