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Chapter S6 Elementary probability Learning Objectives – Understand elementary probability concepts – Calculate the probability of events – Distinguish between mutually exclusive, dependent and independent events – Calculate conditional probabilities – Understand and use the general addition law for probabilities – Understand and apply Venn diagrams – Understand and apply probability tree diagrams © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 1 1 Probability of events Sample space – When a statistical experiment is conducted, there are a number of possible outcomes. These are called sample space and are often denoted by S A coin is tossed. The sample space is: S = {head, tail} © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 2 2 Definitions for probability of events An event is some subset of a sample space. A coin is tossed. Define an event A to be: A = outcome is a head © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 3 3 Definitions for probability of events The impossible event (or empty set) is one that contains no outcomes. It is often denoted by the Greek letter (phi). © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 4 4 Impossible event Example: A hand of 5 cards is dealt from a deck. Let A be the event that the hand contains 5 aces. Since there are only 4 aces in the deck, event A cannot occur. Hence A is an impossible event. © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 5 5 Definitions for probability of events If A is an event, the probability that it occurs is denoted by P(A). The probability (or chance) that an event A occurs is the proportion of possible outcomes in the sample that yield the event A. That is: P A Number of outcomes that yield event A Total number of possible outcomes © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 6 6 Definitions for probability of events Two events A and B are said to be mutually exclusive if they cannot occur simultaneously A = outcome is a head B = outcome is a tail Since A and B cannot both occur, the events are mutually exclusive. © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 7 7 Definitions for probability of events Suppose that A1, A2, A3…An are n mutually exclusive events then: P(A1 or A2…or An) = P(A1) + P(A2) + …+ P(An) © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 8 8 Definitions for probability of events Two events A and B are independent if the occurrence of one does not alter the likelihood of the other event occurring. Events that are not independent are called dependent. © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 9 9 Definitions for probability of events Suppose that A1, A2, A3…An are n independent events then: P(A1) and P(A2)…and P(An) = P(A1) × P(A2) ×…P(An) © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 10 10 Definitions for probability of events The complements of an event are those outcomes of a sample space for which the event does not occur. Two events that are complements of each other are said to be complementary © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 11 11 Definitions for probability of events The probability that event A occurs, given that an event B has occurred, is called the conditional probability that A occurs given that B occurs. The notation for this conditional probability is P(AB). For any two events, A and B, the following relationship holds: P A and B P A B PB © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 12 12 General addition law When two events are not mutually exclusive we should use the following general additional law: P(A or B) = P(A) + P(B) - P(A and B) Note: If the events A and B are mutually exclusive, P(A and B) = 0. © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 13 13 Venn diagrams Sample spaces and events are often presented in a visual display called a Venn diagram. While there are several variations as to how these diagrams are drawn, we will use the following conventions. 1. A sample space is represented by a rectangle. 2. Events are represented by regions within the rectangle. This is usually done using circles (or parts of circles). © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 14 14 Venn diagrams © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 15 15 Venn diagrams The union of two events A and B is the set of all outcomes that are in event A or event B. The notation is: Union of event A and event B A B © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 16 16 Venn diagrams The intersection of two events A and B is the set of all outcomes that are in both event A and event B. The notation is: Intersecti on of event A and event B A B © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 17 17 Probability tree diagrams A visual display of the probabilities using a probability tree diagram. Especially useful for determining probabilities involving events that are not independent. Conditional probabilities are the probabilities on the second tier of branches. © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 18 18 Probability tree diagrams © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher Slide 19 19