Business Statistics - QBM117

1. Business Statistics - QBM117 Assigning probabilities to events

2. Objectives • To define probability; • To describe the relationship between randomness and probability; • To define a random experiment; • To revise the methods of assigning probabilities; • To introduce some of the probability rules.

3. What is probability? • Probability is a numerical measure of uncertainty; • It is a number that conveys the strength of our belief in the occurrence of an uncertain event; • It is often associated with gambling; • It is now an indispensable tool in the analysis of situations which involve uncertainty.

4. Randomness • What is the mean age of all the students in this class? • From a random sample of 10 students we can estimate the mean age of all the students. • How can the mean, based on only a sample of 10, be an accurate estimate of the population mean, ? • A second random sample would most likely produce a different value for the mean.

5. This is due to sampling variability. • Why is this not a problem? • Chance behaviour is unpredictable in the short term but has a regular and predictable pattern in the long term. • For example, consider the experiment of tossing a coin • The results cannot be predicted in advance but there is a pattern which emerges, only after repeated sampling. • This is the basis for probability.

6. Randomness and probability • A phenomenon is called random if the individual outcomes are uncertain, but there is a regular distribution of outcomes in a large number of repetitions. • The probabilityof any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. Probability is an idealisation based on what would happen in an infinitely long series of trials.

7. Random experiments We begin our study of probability by considering the random experiment, as this process generates the uncertain outcomes to which we assign probabilities. Arandom experimentis any well defined procedure that results in one of a number of possible outcomes. The outcome that occurs, cannot be predicted with certainty. For example, rolling a die and observing the number uppermost on the die.

8. Important • The actual outcome of a random experiment cannot be determined in advance. • We can only talk about the probability that a particular outcome will occur.

9. The sample space is the complete list of all the possible outcomes of an experiment. Example If we roll a die, the uppermost face can be a 1, 2, 3, 4, 5 or a 6. These numbers form the sample space. 5 1 2 4 3 6 Sample Space

10. Events The desired outcome or outcomes from the sample space. Example Rolling a die with an even number uppermost can be an event. Sample Space 1 3 5 2 4 6

11. Example In a family of two children the event of there being at least one boy is: Sample Space girl / girl boy / boy girl / boy boy / girl

12. Notation Events can be described in written form, provided the label is defined before hand. Example If we roll a die the uppermost face can be either a 1, 2, 3, 4, 5 or a 6. These numbers form the sample space. The sample space can be written as:

13. Example When rolling a die, let A equal the even numbers. When rolling a die, let B equal the numbers less than three.

14. Probability of an event The probability of an event is defined as thenumber of members in an eventdivided by thenumber of members in the sample space. Where n(A) is the number of members in the event A and n (S ) is the number of members in the sample space.

15. Example When rolling a die: {The event of an even number} The probability of A is:

16. Example On a particular day a statistics lecture has 260 students attending. There are 65 mature aged students in the lecture. If one student is selected at random, what is the probability that a mature age student is selected? S ={all the students in the lecture} M = {mature age students in the lecture}

17. Complement of an event The complement of an event are those members of the sample space that are not contained within the event. Complement rule

18. Example When rolling a die, event A is defined as the even numbers. Find : Sample Space 1 3 5 A 2 4 6

19. Intersection The word used to represent the intersection of two events is: AND The intersection of two events are the members that are common to both events.

20. Example When rolling a die, let A be the event an even number is uppermost. When rolling a die, let B be the event a number less than three is uppermost.

21. It’s just me, all alone… Find the intersection of A and B. Sample Space What number/s are common to both A and B? 1 3 5 2 4 6

22. Find the probability of A and B.

23. Put your hand up if you look like me! Example Let A be defined as the students with 2 blue eyes in a lecture theatre and B be defined as the students with two brown eyes in the lecture theatre. Then the probability students will have two blue and two brown eyes in the lecture theatre is ...

24. Mutually exclusive events Two events are mutually exclusive when they have no members in common. (They don’t share any members) Example A is the event an even number is uppermost on a die B is the event an odd number is uppermost on die A B 1 3 5 2 4 6

25. Union of events The word used to represent the union of two events is: OR The union of two events are those members that are in one event or the other event or in both.

26. Example When rolling a die, let X be the event an odd number is uppermost. When rolling a die, let Y be the event a number less than five is uppermost.

27. All except for me !! Find the probability of X or Y. What number/s are either in X or in Y? Sample Space 1 3 5 2 4 6

28. Therefore the probability of X or Y is

29. Addition formula We do not always know the members of each event, rather we only know the probabilities of these events. In such cases, there is a formula that enables us to find the probability of A or B:

30. Intersection Addition formula Sample Space A A B B Counted once with P(A) Counted twice with P(B)

31. Example In a population the probability of being female is 0.6 and the probability of being aged 30 and over is 0.4. The probability of being female and aged 30 and over is 0.2. Find the probability of being either female or aged 30 and over. Let F represent the females in the population Let O represent the people aged 30 and over

32. Example In a population the probability of being female is 0.6and the probability of being aged 30 and over is 0.4. The probability of being female and aged 30 and over is 0.2. Find the probability of being either female or aged 30 and over.

33. Reading for next lecture Chapter 4 sections 4.4 - 4.5 Exercise to be completed before next lecture S&S 4.5 4.7 4.13 4.15 4.69