1. EXAMPLE 1 Find probabilities of events You roll a standard six-sided die. Find the probability of (a) rolling a 5 and (b) rolling an even number. SOLUTION a. There are 6 possible outcomes. Only 1 outcome corresponds to rolling a 5. Number of ways to roll the die P( rolling a 5) = Number of ways to roll a 5 = 1 6

2. EXAMPLE 1 Find probabilities of events b. A total of 3 outcomes correspond to rolling an even number: a 2, 4, or 6. P( rolling even number ) = Number of ways to roll an even number Number of ways to roll the die = 3 6 1 2 =

3. EXAMPLE 2 Use permutations or combinations Entertainment A community center hosts a talent contest for local musicians. On a given evening, 7 musicians are scheduled to perform. The order in which the musicians perform is randomly selected during the show. a. What is the probability that the musicians perform in alphabetical order by their last names? (Assume that no two musicians have the same last name.)

4. EXAMPLE 2 Use permutations or combinations SOLUTION a. There are 7! different permutations of the 7 musicians. Of these, only 1 is in alphabetical order by last name. So, the probability is: P( alphabetical order ) = 1 7!7! 1 5040 = ≈ 0.000198

5. EXAMPLE 2 Use permutations or combinations b. There are 7 C 2 different combinations of 2 musicians. Of these, 4 C 2 are 2 of your friends. So, the probability is: P( first 2 performers are your friends ) = 4C24C2 7C27C2 6 21 = 0.286 2 7 =

6. GUIDED PRACTICE for Examples 1 and 2 You have an equally likely chance of choosing any integer from 1 through 20. Find the probability of the given event. 1. A perfect square is chosen. SOLUTION Number of Integers P = Number of perfect squares = 1 5 4 20 =

7. GUIDED PRACTICE for Examples 1 and 2 You have an equally likely chance of choosing any integer from 1 through 20. Find the probability of the given event. 2. A factor of 30 is chosen. ANSWER 7 20 SOLUTION Number of Integers P = Factors of 30 between 1 and 20 7 20 =

8. GUIDED PRACTICE for Examples 1 and 2 What If? In Example 2, how do your answers to parts (a) and (b) change if there are 9 musicians scheduled to perform? 3. SOLUTION a. There are 9! different permutations of the 9 musicians. Of these, only 1 is in alphabetical order by last name. So, the probability is: P( alphabetical order ) = 1 9! 1 362,880 = ≈ 0.0000027 The probability would decrease to ANSWER 1 362,880

9. b. There are 9 C 2 different combinations of 2 musicians. Of these, 4 C 2 are 2 of your friends. So, the probability is: P( first 2 performers are your friends ) = 4C24C2 9C29C2 6 36 = 1 6 = GUIDED PRACTICE for Examples 1 and 2 The probability would decrease to ANSWER 1 6