Title: Lesson 1–1
Author: Teri Willard
Last saved by: Glencoe Mathematics

Lesson 12–1

Example 1  Independent Events
RECREATION  At a local ski resort you can rent ski equipment. At the rental shop, you can choose standard, deluxe, or professional skis with economy, regular, or insulated ski boots. Any type of boot can be adjusted to fit any type of skis. How many different combinations of skis and boots are available?

First, note that the choice of the type of boots does not affect the choice of the type of skis, so these events are independent.

Method 1  Tree Diagram 
S represents standard skis, D deluxe skis, P professional skis, E economy boots, R regular boots, and I insulated boots. 

Method 2  Make a table
Make a table in which each row represents a type of skis and each column represents a type of boot.

| | | Boots |
| | Economy | Regular | Insulated
| Standard | SE | SR | SI
Skis | Deluxe | DE | DR | DI
| Professional | PE | PR | PI

There are nine possible outcomes.

Example 2  Fundamental Counting Principle
Multiple–Choice Test Item

BUSINESS  Miriam has a home–based business making gift baskets. One particular basket for children contains one package of snacks and one stuffed toy. For a customized basket, you can select one of five different snack packages and one of seven different stuffed toys. How many different children’s gift baskets are possible?

A 35                               B  12                                 C  24                             D  18

Read the Test Item
The choice of a type of snack package does not affect the choice of a stuffed toy, so these events are independent.

Solve the Test Item
There are 5 ways to choose a snack package and 7 ways to choose a stuffed toy. By the Fundamental Counting Principle, there are 5  7 or 35 possible gift baskets. The answer is A.

Example 3  More than Two Independent Events
SCHOOL  Brad is planning to order a class ring. He has choices in the metal option, the stone option, and the initial option. For the metal option, he can order either silver or gold. For the stone option, Brad can choose to have no stone, a stone in the color of his birthstone, or a stone in special school colors. Finally, for the initial option, he can choose to have no initials, his initials, the school’s initials, or a special sport’s insignia inscribed on the ring. How many different choices for a ring does Brad have?

The choice of any of the three options does not affect the choice of the other options, so the choices in each option are independent events.

There are 2 metal options, 3 stone options, and 4 initial options. There are 2 • 3 • 4 or 24 possible different rings.

Example 4  Dependent Events
MUSIC  Aislyn is having a birthday party and plans to play her 8 favorite CDs. She does not want to repeat playing any of the CDs. How many different ways can she play the 8 CDs?

When Aislyn picks a particular CD to play, she cannot play it again since she does not want to repeat any of the CDs. Therefore, the choices of which CD to play are dependent events.

There are 8 CDs that Aislyn can play first. That leaves 7 CDs for her to choose to play second. After she chooses two CDs, there are 6 CDs to choose from, and so on.

Order of play | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | 8th
Number of Choices | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1

There are 8  7  6  5  4  3  2  1 or 40,320 ways that she could play the CDs. Note that 
8  7  6  5  4  3  2  1 = 8!.