1 Discrete Structures - CSIS2070 Text Discrete Mathematics and Its Applications Kenneth H. Rosen Chapter 4 Counting

2 Section 4.4 Discrete Probability

3 Finite Probability The probability of an even E, which is a subset of a finite sample space S of equally likely outcomes, is

4 Example Suppose two dice are rolled. The sample space would be x x x x x x 2 x x x x x x 3 x x x x x x 4 x x x x x x 5 x x x x x x 6 x x x x x x

5 p(sum is 11) |S| = 36

x x x x x x 2 x x x x x x 3 x x x x x x 4 x x x x x x 5 x x x x x x 6 x x x x x x p(sum is 11) |S| = 36

x x x x x x 2 x x x x x x 3 x x x x x x 4 x x x x x x 5 x x x x x x 6 x x x x x x p(sum is 11) |S| = 36 |E| = 2

8 Suppose a lottery randomly selects 6 numbers from 40. What is the probability that you selected the correct six numbers? Order is not important |E| = 1 |S| = C(40,6) p(E) = 1 C(40,6)

9 Combinations of Events Let E be an event in a sample space S. The probability of the event E, the complementary event of E, is given by p(E) = 1 - p(E)

10 Combinations of Events Let E 1 and E 2 be events in the sample space S. Then

11 Example Suppose a red die and a blue die are rolled. The sample space would be x x x x x x 2 x x x x x x 3 x x x x x x 4 x x x x x x 5 x x x x x x 6 x x x x x x

12 p(sum is 7 or blue die is 3) |S| = x x x x x x 2 x x x x x x 3 x x x x x x 4 x x x x x x 5 x x x x x x 6 x x x x x x

13 p(sum is 7 or blue die is 3) |S| = x x x x x x 2 x x x x x x 3 x x x x x x 4 x x x x x x 5 x x x x x x 6 x x x x x x |sum is 7| = 6 |blue die is 3| = 6 | in intersection | = 1 p(sum is 7 or blue die is 3) = 6/36 + 6/36 - 1/36 = 11/36

14 Probability of four-of-a-kind, dealing five Aces Kings Queens Jacks tens nines eights sevens sixes fives fours threes twos

15 finished