Counting Outcomes Day 2 The Fundamental Counting Principle

Counting Outcomes (Day 2) The Fundamental Counting Principle 1

Using a table The table below shows all of the possible outcomes for rolling two number cubes. Going across the top are the outcomes for rolling the first number cube. Going down the left are all the outcomes for rolling the second number cube. Shade all of the possible outcomes that are doubles. The probability of rolling doubles is or 1 2 3 4 5 6 1 (1, 1) (2, 1) (3, 1) (4, 1) (5, 1) (6, 1) 2 (1, 2) (2, 2) (3, 2) (4, 2) (5, 2) (6, 2) 3 (1, 3) (2, 3) (3, 3) (4, 3) (5, 3) (6, 3) 4 (1, 4) (2, 4) (3, 4) (4, 4) (5, 4) (6, 4) 5 (1, 5) (2, 5) (3, 5) (4, 5) (5, 5) (6, 5) 6 (1, 6) (2, 6) (3, 6) (4, 6) (5, 6) (6, 6) 2

Make your own table. • Make a table showing all of the possibly outcomes for flipping two integer chips. Each integer chip has one side that is red, and one side that is yellow. • What is the probability of flipping the chips and getting the same color? Red Yellow Red (red, red) (red, yellow) Yellow (yellow, red) (yellow, yellow) 3

Fundamental Counting Principle • If event M can occur in m ways and is followed by event N that can occur in n ways, then the event M followed by the event N can occur m(n) ways. – Example: If a number cube is rolled and a coin is tossed, there are 6(2) outcomes, or 12 possible outcomes. • The fundamental counting principle can be used to find the total number of outcomes possible when 2 or more events occur. • It does not create a list of the possible outcomes. 4

Example 1 • In the United States, radio and television stations use call letters that start with K or W. How many different call letters with four letters are possible? Use the fundamental counting principle to solve this question. Number of Total number possible letters X possible letters = of possible call for the first for the second for the third for the fourth letters letter 2 X 26 There are 35, 152 possible call letters for radio and tv stations. = 35, 152 5

Example 2 • What is the probability of winning a lottery game where the winning number is made up of three digits from 0 to 9 chosen at random? Use the fundamental counting principle to find the number of possible outcomes, and write the probability as a fraction. 10 × 10 = 1, 000 There are 1, 000 possible outcomes. However, there is only 1 winning number. The probability of winning with one ticket is . 6

• Practice 1: At a pizza parlor, you can choose from 3 types of crust, 2 types of cheese, and 4 toppings. How many different one-cheese and one-topping pizzas can be ordered? Crusts Cheese Toppings 3(2)(4)=24 Regular Deep dish Thin Mozzarella Feta Pepperoni Sausage Mushroom Peppers There are 24 different one cheese, and one topping pizzas that can be ordered. • Practice 2: A map shows three towns. There are 4 roads between towns A and B, and there are 5 roads between towns B and C. How many different routes can you travel from town A to town B to town C? 4 (5) =20 There are 20 different routes you can travel. 7

• Practice 3: Find the number of possible choices for a 2 -digit number that is greater than 19. The first number bigger than 19 is 20. 8 (10) =80 The first digit has to be greater than 2, so there are 8 numbers between 0 -9 that are greater than 2 Since 20 is greater than 19, the second digit in the number can be numbers all of the numbers 0 -9. There are 80 twodigit numbers that are greater than 19. • Practice 4: Find the number of possible choices for a 4 -digit PIN number if the digits cannot be repeated. 10 (9)(8) (7)=5, 040 The second digit The first digit has to be one less number, because can be any number 0 -9. the pin can’t repeat. There are 5, 040 possible choices for a 4 -digit PIN number where the digits don’t repeat. The third digit has to be one less than the second, because the pin can’t repeat. The last digit has to be one less than the third, because the pin can’t 8 repeat.