Chapter 7 Sets & Probability Section 7.3 Introduction to Probability

Why use probability? A great many problems that come up in the applications of mathematics involve random phenomena – those for which exact prediction is impossible. The best we can do is determine the probability of the possible outcomes.

Sample Spaces In probability, an experiment is an activity or occurrence with an observable result. Each repetition of an experiment is called a trial. The possible results of each trial are called outcomes. The set of all possible outcomes for an experiment is the sample space for that experiment.

Example: A spinner that is equally divided up into three spaces is spun and a coin is tossed. Give the sample space for this experiment. SpinnerCoin TossPossible Outcomes H(1, H) H(1, H) 1 T (1, T) T (1, T) H (2, H) H (2, H) 2 T (2, T) T (2, T) H (3, H) H (3, H) 3 T (3, T) T (3, T) S = { (1,H), (1,T), (2,H), (2,T), (3,H), (3,T) } S = { (1,H), (1,T), (2,H), (2,T), (3,H), (3,T) }

Events An event is a subset of a sample space. An event in which only one outcome is possible is called a simple event. If the event equals the sample space, then the event is called a certain event. If the event is equal to the null, or empty, set, then the event is called an impossible event.

Example: Consider rolling a single die. S = { 1, 2, 3, 4, 5, 6} S = { 1, 2, 3, 4, 5, 6} Event A: Rolling a 5 Event A: Rolling a 5 Event B: Rolling an odd number Event B: Rolling an odd number Event C: Rolling a number less than 7 Event C: Rolling a number less than 7 Event D: Rolling a number greater than 6 Event D: Rolling a number greater than 6 Which event is a simple event? Event A Which event is an impossible event? Event D Which event is a certain event? Event C

Set Operations for Events Let E and F be events for a sample space, S. E  F occurs when both E and F occur; E  F occurs when both E and F occur; E  F occurs when E or F or both occur; E  F occurs when E or F or both occur; E ′ occurs when E does not occur. E ′ occurs when E does not occur. Mutually Exclusive Events Events E and F are mutually exclusive events if E  F = Ø

Probability For sample spaces with equally likely outcomes, the probability of an event is defined as follows. Basic Probability Principle: Let S be a sample space of equally likely outcomes, and let event E be a subset of S. Then the probability that event E occurs is For any event E, 0 ≤ P(E) ≤ 1. For any event E, 0 ≤ P(E) ≤ 1.

Example: A marble is drawn from a bowl containing 3 yellow, 4 white, and 8 blue marbles. Find the probability of the following events. 1.) A yellow marble is drawn P (yellow) = 3 / 15 = 1 / 5 P (yellow) = 3 / 15 = 1 / 5 2.) A blue marble is drawn P (blue) = 8 / 15 P (blue) = 8 / 15 3.) A white marble is not drawn P (not white) = 11 / 15 P (not white) = 11 / 15