15-1 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Chapter 15 Elementary Probability Introductory Mathematics & Statistics

15-2 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 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

15-3 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.1 Introduction In everyday language we often refer to the probability that certain events will happen We also use the word ‘chance’ as a substitute for probability on some occasions While we all use the word ‘probability’ in our language, there would be few people who could provide a formal definition of its meaning Examples –There is a 10% chance that it will rain –There is a 30% chance that Essendon will win the AFL premiership in the year 2010 –There is a 25% chance that a certain investment will yield a profit in the coming year –There is a 50–50 chance that I will get a tax refund next year –The probability that a 767 jet plane will crash into the Sydney Harbour Bridge before the year 2030 is 1 in 100 million

15-4 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.2 Probability of events Sample space –When a statistical experiment is conducted, there are a number of possible outcomes –These possible outcomes are called a sample space and this is denoted by S  E.g. a coin is tossed. What is the sample space?  Solution: S = {head, tail} Events –An event is a specified subset of a sample space.  E.g. a coin is tossed. Define event A as the outcome ‘heads’  Solution: A = outcome is a head

15-5 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.2 Probability of events (cont…) Events (cont…) –More than one event can be defined from a sample space.  E.g. suppose a card is drawn at random from a pack of 52 playing cards. Define events A, B and C as drawing an ace, red card and face card, respectively  Solution: A = card drawn is an ace, B = card drawn is red, C = card drawn is a face card –The impossible event (or empty set) is one that contains no outcomes. It is often denoted by the Greek letter (phi)  E.g. a hand of 5 cards is dealt from a deck z. Let A be the event that the hand contains 5 aces. Is this possible?  Solution : Since there are only 4 aces in the deck, event A cannot occur. Hence A is an impossible event.

15-6 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.2 Probability of events (cont…) Probability –If A is an event, the probability that it will occur is denoted by P(A) –The probability (or chance) that an event A will occur is the proportion of possible outcomes in the sample space that yield the event A. That is: –The definition makes sense only if the number of possible outcomes (the sample space) is finite –If an event can never occur, its probability is 0. An event that always happens has probability 1 –The value of a probability must always lie between 0 and 1 –A probability may be expressed as a decimal or a fraction

15-7 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.2 Probability of events (cont…) Mutually exclusive events –Two events A and B are said to be mutually exclusive if they cannot occur simultaneously –If two events A and B are mutually exclusive, the following relationship holds: –Suppose that are mutually exclusive events. Then:

15-8 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.2 Probability of events (cont…) Independent events –Two events A and B are independent events if the occurrence of one does not alter the likelihood of the other event occurring –Events that are not independent are called dependent events –If two events A and B are independent, the following relationship holds: –Suppose that are n independents events. Then

15-9 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.2 Probability of events (cont…) Complementary events –The complement of an event is the set of 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 events (Note: complementary events are mutually exclusive) –Suppose we define the events: A = no one has the characteristic B = at least 1 person has the characteristic Then A and B are complementary events P (at least 1 person has the characteristic) = 1 – P (no person has the characteristic)

15-10 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.2 Probability of events (cont…) Conditional probabilities –The probability that event A will occur, given that an event B has occurred, is called the conditional probability that A will occur, given that B has occurred –The notation for this conditional probability is P (A|B) –For any two events, A and B, the following relationship holds:

15-11 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.2 Probability of events (cont…) Conditional probabilities (cont…)Conditional probabilities (cont…) –If two events A and B are independent –Substituting this result –That is, for independent events A and B the conditional probability that event A will occur, given that event B had occurred, is simply the probability that event A will occur

15-12 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.2 Probability of events (cont…) The general addition law –When two events are not mutually exclusive, use the following general addition law –If the events A and B are mutually exclusive, P(A and B) = 0

15-13 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.3 Venn diagrams Sample spaces and events are often presented in a visual display called a Venn diagram Use the following conventions –A sample space is represented by a rectangle –Events are represented by regions within the rectangle. This is usually done using circles Venn diagrams are used to assist in presenting a picture of the union and intersection of events, and in the calculation of probabilities

15-14 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.3 Venn diagrams (cont…) Definitions –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 Hence, we could write, for example, P (A ∪ B) instead of P(A or B) –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: Intersection of event A and event B = A ∩ B Hence, we could write, for example, P (A ∩ B) instead of P(AandB)

15-15 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.3 Venn diagrams (cont…) The shaded area is event A

15-16 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.3 Venn diagrams (cont…) The union of two events A and B is the set of all outcomes that are in event A or event B

15-17 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.3 Venn diagrams (cont…) The intersection of two events A and B is the set of all outcomes that are in both event A and event B.

15-18 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.3 Venn diagrams (cont…) The intersection of events A, B and C is the set of all outcomes that is in events A, B and C A B C

15-19 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.4 Probability tree diagrams Probability tree diagrams can be a useful visual display of probabilities The diagrams are especially useful for determining probabilities involving events that are not independent The joint probabilities for combinations of these events are found by multiplying the probabilities along the branches from the beginning of the tree If the events are not independent, the probabilities on the second tier of branches will be conditional probabilities, since their values will depend on what happened in the first event

15-20 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.4 Probability tree diagrams (cont..) Example –A clothing store has just imported a new range of suede jackets that it has advertised at a bargain price on a rack inside the store. The probability that a customer will try on a jacket is If a customer tries on a jacket, the probability that he or she will buy it is If a customer does not try on a jacket, the probability that he or she will buy it is –Calculate the probability that: (a) a customer will try on a jacket and will buy it (b) a customer will try on a jacket and will not buy it (c) a customer will not try on a jacket and will buy it (d) a customer will not try on a jacket and will not buy it

15-21 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 15.4 Probability tree diagrams (cont..) Solution

15-22 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Summary We have looked at understanding elementary probability concepts We calculated the probability of events We distinguished between mutually exclusive, dependent and independent events We also looked at calculating conditional probabilities We understood and used the general addition law for probabilities We understood and applied Venn diagrams We understood and applied probability tree diagrams