(figure available in print form)
The chart above lists all the ways that two dice can come up. A list of all the ways something can end up is called the
Sample Space
for that game or experiment or whatever.
The probability of something is the ratio of the number of ways the thing can happen to the number of things that can happen. Let us find the probability of a one on the first die, P(one on first die). How many points in the table have a one come first? ___ Put a box around them. How many points are there altogether? ___ So the ratio of number of ways to have one on first die to number of ways the dice can come up is 6 to 36, which we reduce to 1/6. So P (one on first die) = 1
6
So how do we find the probability of an event? First make your sample space, the list of everything that can happen. Then circle the set of things you want to happen. Count the number of things in your set, count the number of things in the whole sample space. The ratio of the first number to the second number, what I want over what can happen, is my probability. It is a fraction so reduce it if possible.
How would you describe the points in the upside down T box? So what is the probability of getting a 6 on the second die or a 3 on the first? How many points are in the box? How many points are there altogether? So P(6 on first OR 3 on second) is 11/36.
How would you describe the event in the oval? So what is the probability of getting a 3 on the first die and a 6 on the second? P(3 on first die AND 6 on second die) = 1/36.
Do people usually say “I have a 3, on my first die and a 6 on my second?” Of course not. What do they say? They add the 3 and the 6 and say: “I have a nine.” So make a new sample space chart, but this time instead of (1,1) put 2 and instead of (2,3) put 5 and so forth for all 36 points.
Use your new sample space to find these probabilities.
P(2), P(3), P(4), P(5), P(6), P(7), P(8), P(9), P(10), P(11), P(12).
Make a bar graph of your probabilities from above. Have the scores from 2 to 12 across the bottom and the probabilities go up the side.
Multiplying Polynomials
Here is a way to change the dice sample space into a multiplication problem. First let us look at a different way to multiply two binomials. (2a + 3)(5a + 4). First make a two by two square with 2a + 3 across the top and 5a + 4 along the side. Each of the boxes is filled in by multiplying the term at the top by the term at the side. Then combine like terms.
(figure available in print form)
We get the same answer as normally.5 Now6use this square table method to multiply x + x
2
+ x
3
+ x
4
+ x
5
+ x
6
by itself.
(figure available in print form)
Look at the chart, if the x’s were erased inside the boxes what would it look like? Look at your sample space for throwing two dice. How many ways can you throw 7 with two dice? ___ What is the coefficient on x7 in the expansion? ___ So what do the exponents stand for? What do the coefficients stand for? ___
This is the idea of generating functions. Since computers can multiply polynomials they can calculate probabilities if they have the correct generating function. So the generating function for the number of ways two dice can come up is (x + x
2
+ x3 + x
4
+ x
5
+ x
6
)
2
, where the coefficient on xn tells the number of ways n can be thrown.
In fact if you let x = 1 in the generating function you will find the total number of ways the dice can come up. (1 + 1 + 1 + 1 + 1 + 1)
2
= (6)
2
= 36.
The point of this activity is to show how math is interconnected. Ideas in one course appear in another. I certainly did not claim I was teaching generating functions. Only that generating functions and multiplying polynomials can be mentioned at an early date. This is part of a math course called Combinatorics. Some people call combinatorics counting without counting. Look at the generating function, it counts the number of ways the dice can come up, but we did not actually count 1,2,3 to get the answers.
