MATH 1040 - Introduction to Statistics

Lesson 11.1 Fundamental Counting Rule

Reading

Reading sections are from the Introductory Statistics Textbook

• 7.1.1 —

Lesson

We have discussed the probabilities of a single event (lessons 9-10). But what if you have multiple events at the same time? For example, we roll not one die but two dice. Or instead of choosing marbles from 5 different bags. Or choosing toppings for a pizza (choosing meats, cheese, and toppings). All of these have multiple events. In this lesson, we will learn how to calculate the probabilities of multiple events.

Consider two dice - one 4-sided white die and one 6-sided black die. If the white die rolls a 3, we’ll call that W3, and if the black die rolls a 5, we’ll call that B5. Now, roll the dice. How many possibilities are there?

• If we roll W1, there are 6 possibilities for black: (W1,B1),(W1,B2),(W1,B3),(W1,B4),(W1,B5),(W1,B6)
• If we roll W2, there are 6 possibilities for black: (W2,B1),(W2,B2),(W2,B3),(W2,B4),(W2,B5),(W2,B6)
• If we roll W3, there are 6 possibilities for black: (W3,B1),(W3,B2),(W3,B3),(W3,B4),(W3,B5),(W3,B6)
• If we roll W4, there are 6 possibilities for black: (W4,B1),(W4,B2),(W4,B3),(W4,B4),(W4,B5),(W4,B6)
• Bottom line: There are 4 possibilities for the white die. For each possibility, there are 6 possibilities for the black die. That is a total of \(4\times 6 = 24\) possibilities

That’s it! That is the Fundamental Counting Rule.

If event A has \(k\) different possibilities, and event B has \(n\) different possibilities, then the total number of combined possibilities is \(k\times n\).

What about 3 events? Let’s take the 4-sided white die and the 6-sided black die, and we’ll add a 10-sided red die.

• There are 24 possibilities with the white and black dice
• For each possibility with the white and black dice, there are 10 possibilities for the red die
• That’s a total of \(24\times 10 = 240\) possible combinations

The complete Fundamental Counting Rule.

For multiple events, let’s say that event \(i\) has \(n_i\) different possible outcomes. The total possible combinations is, \(n_1 \times n_2 \times n_3 \times \dots\).

Practice

Here are three example problems using the Fundamental Counting Rule. Answer the questions to the best of your ability then afterwards check the answer.

Practice Problem 11.1.1

Papa Miguel’s pizza place allows you to choose your sauce, cheese and 1 topping for a flat price. The menu has the following options:

• Sauce: Tomato, BBQ, Alfredo
• Cheese: Mozerella, 5-cheese, Mexican, Parmesan
• Toppings: Pepperoni, Sausage, Chicken, Ham, Bacon, Pineapple, Onion, Taco Beef

How many possible pizza combinations could you create?

• After solving on your own, check the solution.

Practice Problem 11.1.2

There are 13 people in a race. How many ways can the 13 runners cross the finish line?

• After solving on your own, check the solution.

Practice Problem 11.1.3

If you have a secure password, then it is very difficult for a hacker to randomly guess your password. Is it safer to have an 8-character password that can contain any character {a-z, A-Z, 0-9, !@#$%^&*() } or a 12-character password with only lower-case letters?

Take a second and make a guess as to which is more secure. Then use the Fundamental Counting Rule to find the number of possible 8-character and 12-character passwords. The option with more possible passwords would be more secure as it is more difficult to randomly guess.

• After solving on your own, check the solution.