15-1 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher,...

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

Chapter 15

Elementary Probability

Introductory Mathematics & Statistics


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


15.1 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


15.2 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


15.2 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.


15.2 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

outcomespossibleofnumbertotal

AeventyieldthatoutcomesofnumberAP


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:

BPAPBorAP

n321 A,A,A,A

n321

n321

APAPAPAP

AAorAorAP



• 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

n321

n321

APAPAPAP

AandAandAandAP

BPAPAandBP

n321 A,A,A,A



• 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 characteristicB = 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)



• 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:

BP

BandAPBAP


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

BPAPBAP

AP

BP

BPAPBAP



• 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

BandAPBPAPBorAP


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


15.3 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)


15.3 Venn diagrams (cont…)

The shaded area is event A



• The union of two events A and B is the set of all outcomes that are in event A or event B

BABA eventandeventofUnion



• The intersection of two events A and B is the set of all outcomes that are in both event A and event B.

BABA eventandeventofonIntersecti



• The intersection of events A, B and C is the set of all outcomes that is in events A, B and C

CBACandBandAeventsofonIntersecti

A B

C


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


15.4 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 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


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

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