Slide 1 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 1 n 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 Elementary probability Chapter S6

Slide 2 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 2 Probability of events n Sample space sample space S –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}

Slide 3 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 3 Definitions for probability of events A coin is tossed. Define an event A to be : A = outcome is a head event An event is some subset of a sample space.

Slide 4 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 4 Definitions for probability of events impossible event The impossible event (or empty set) is one that contains no outcomes. It is often denoted by the Greek letter (phi).

Slide 5 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 5 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.

Slide 6 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 6 Definitions for probability of events 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 ). If A is an event, the probability that it occurs is denoted by P ( A ).

Slide 7 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 7 Definitions for probability of events A A = outcome is a head B B = outcome is a tail Since A and B cannot both occur, the events are mutually exclusive. mutually exclusive Two events A and B are said to be mutually exclusive if they cannot occur simultaneously

Slide 8 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 8 Definitions for probability of events n mutually exclusive Suppose that A 1, A 2, A 3 … A n are n mutually exclusive events then: P(A 1 or A 2 …or A n ) = P(A 1 ) + P(A 2 ) + …+ P(A n )

Slide 9 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 9 Definitions for probability of events independent dependent. 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.

Slide 10 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 10 Definitions for probability of events Suppose that A 1, A 2, A 3 … A n are n independent events then: P(A 1 ) and P(A 2 )…and P(A n ) = P(A 1 ) × P(A 2 ) ×…P(A n )

Slide 11 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 11 Definitions for probability of events complements not The complements of an event are those outcomes of a sample space for which the event does not occur. complementary Two events that are complements of each other are said to be complementary

Slide 12 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 12 Definitions for probability of events conditional probability P ( A  B ). 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 ( A  B ). For any two events, A and B, the following relationship holds:

Slide 13 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 13 General addition law general additional 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.

Slide 14 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 14 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. sample space 1. A sample space is represented by a rectangle. Events circles 2. Events are represented by regions within the rectangle. This is usually done using circles (or parts of circles).

Slide 15 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 15 Venn diagrams

Slide 16 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 16 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:

Slide 17 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 17 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:

Slide 18 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 18 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 Conditional probabilities are the probabilities on the second tier of branches.

Slide 19 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 19 Probability tree diagrams