Examples:
1. Activities: roll a die and flip a coin
There are 6 ways to roll a die and 2 ways to flip a coin.
There are 6 • 2 = 12 ways to roll a die and flip a coin.
(The sample space will contain a count of 12 possible outcomes.)
2. Activities: draw two cards from a standard deck of 52 cards without replacing the cards
There are 52 ways to draw the first card.
There are 51 ways to draw the second card.
There are 52 • 51 = 2,652 ways to draw the two cards.
(The sample space will contain a count of 2,652 possible outcomes.)
The Counting Principle also works for more than two activities.
3. Activities: a coin is tossed five times
There are 2 ways to flip each coin.
There are 2 • 2 • 2 • 2 •2 = 32 arrangements of heads and tails.
(The sample space will contain a count of 32 possible outcomes.)
4. Activities: a die is rolled four times
There are 6 ways to roll each die.
There are 6 • 6 • 6 • 6 = 1,296 possible outcomes.
(The sample space will contain a count of 1,296 possible outcomes.)
So, when does the Counting Principle NOT work?
5. Activities: You are buying a new skateboard.
There are 2 styles: wood or polymer.
There are 4 colors: red, black, blue, orange.
There are 3 designs: line art, tattoo, splash
There are 2 • 4 • 3 = 24 possible choices of skateboards. Yes, find and dandy!!
BUT... the salesman says he is sorry but they are all sold out of all orange polymer skateboards. Now, how many choices do you have?
Your choices are no longer independent of each other. The color options now depend upon which style board is chosen, wood or polymer.
Well, you still have all of the wood skateboard options which will be 1 • 4 • 3 = 12 choices.
Under the polymer skateboards, you now have 1• 3 • 3 = 9 choices.
This gives a total of
21 choices. |