chapter 15

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

Chapter 15

Elementary ProbabilityIntroductory Mathematics & Statistics

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


Learning ObjectivesUnderstand elementary probability conceptsCalculate the probability of eventsDistinguish between mutually exclusive, dependent and independent eventsCalculate conditional probabilitiesUnderstand and use the general addition law for probabilitiesUnderstand and apply Venn diagramsUnderstand and apply probability tree diagrams



15.1 IntroductionIn everyday language we often refer to the probability that certain events will happenWe also use the word chance as a substitute for probability on some occasionsWhile we all use the word probability in our language, there would be few people who could provide a formal definition of its meaningExamplesThere is a 10% chance that it will rain There is a 30% chance that Essendon will win the AFL premiership in the year 2010There is a 25% chance that a certain investment will yield a profit in the coming yearThere is a 5050 chance that I will get a tax refund next yearThe probability that a 767 jet plane will crash into the Sydney Harbour Bridge before the year 2030 is 1 in 100 million



15.2 Probability of eventsSample spaceWhen a statistical experiment is conducted, there are a number of possible outcomesThese possible outcomes are called a sample space and this is denoted by SE.g. a coin is tossed. What is the sample space?Solution: S = {head, tail}EventsAn event is a specified subset of a sample space.E.g. a coin is tossed. Define event A as the outcome headsSolution: A = outcome is a head



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, respectivelySolution: 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.



15.2 Probability of events (cont)ProbabilityIf 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 1The value of a probability must always lie between 0 and 1A probability may be expressed as a decimal or a fraction



15.2 Probability of events (cont)Mutually exclusive eventsTwo events A and B are said to be mutually exclusive if they cannot occur simultaneouslyIf two events A and B are mutually exclusive, the following relationship holds:

Suppose that are mutually exclusive events. Then:



15.2 Probability of events (cont)Independent eventsTwo events A and B are independent events if the occurrence of one does not alter the likelihood of the other event occurringEvents that are not independent are called dependent eventsIf two events A and B are independent, the following relationship holds:

Suppose that are n independents events. Then



15.2 Probability of events (cont)Complementary eventsThe 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 characteristicB = at least 1 person has the characteristicThen A and B are complementary events P (at least 1 person has the characteristic) = 1 P (no person has the characteristic)



15.2 Probability of events (cont)Conditional probabilitiesThe 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 occurredThe notation for this conditional probability is P (A|B)For any two events, A and B, the following relationship holds:



15.2 Probability of events (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



15.2 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



15.3 Venn diagramsSample spaces and events are often presented in a visual display called a Venn diagram

Use the following conventionsA sample space is represented by a rectangleEvents 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



15.3 Venn diagrams (cont)DefinitionsThe 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)



15.3 Venn diagrams (cont)The shaded area is event A



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



15.3 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.



15.3 Venn diagrams (cont)The intersection of events A, B and C is the set of all outcomes that is in events A, B and CABC



15.4 Probability tree diagramsProbability tree diagrams can be a useful visual display of probabilitiesThe diagrams are especially useful for determining probabilities involving events that are not independentThe joint probabilities for combinations of these events are found by multiplying the probabilities along the branches from the beginning of the treeIf 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



15.4 Probability tree diagrams (cont..)ExampleA 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 0.40. If a customer tries on a jacket, the probability that he or she will buy it is 0.70. If a customer does not try on a jacket, the probability that he or she will buy it is 0.15.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



15.4 Probability tree diagrams (cont..)Solution



SummaryWe have looked at understanding elementary probability conceptsWe calculated the probability of eventsWe distinguished between mutually exclusive, dependent and independent eventsWe also looked at calculating conditional probabilitiesWe understood and used the general addition law for probabilitiesWe understood and applied Venn diagramsWe understood and applied probability tree diagrams


chapter 15

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