1. Copyright © Cengage Learning. All rights reserved. 7 Probability

2. Copyright © Cengage Learning. All rights reserved. 7.4 Probability and Counting Techniques

3. 3 We have seen that, when all outcomes in a sample space are equally likely, we can use the following formula to model the probability of each event: Modeling Probability: Equally Likely Outcomes In an experiment in which all outcomes are equally likely, the probability of an event E is given by

4. 4 Example 1 – Marbles A bag contains four red marbles and two green ones. Upon seeing the bag, Suzan (who has compulsive marble- grabbing tendencies) sticks her hand in and grabs three at random. Find the probability that she will get both green marbles. Solution: According to the formula, we need to know these numbers: The number of elements in the sample space S. The number of elements in the event E.

5. 5 Example 1 – Solution First of all, what is the sample space? The sample space is the set of all possible outcomes, and each outcome consists of a set of three marbles (in Suzan’s hand). So, the set of outcomes is the set of all sets of three marbles chosen from a total of six marbles (four red and two green). Thus, n(S) = C(6, 3) = 20. cont’d

6. 6 Example 1 – Solution Now what about E? This is the event that Suzan gets both green marbles. We must rephrase this as a subset of S in order to deal with it: “E is the collection of sets of three marbles such that one is red and two are green.” Thus, n(E) is the number of such sets, which we determine using a decision algorithm. Step 1 Choose a red marble; C(4, 1) = 4 possible outcomes. Step 2 Choose the two green marbles; C(2, 2) = 1 possible outcome. cont’d

7. 7 Example 1 – Solution We get n(E) = 4  1 = 4. Now, Thus, there is a one in five chance of Suzan’s getting both the green marbles. cont’d