1. 1 Business Statistics - QBM117 Assigning probabilities to events

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

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

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

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

6. 6 Randomness and probability w 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. w The probability of 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. 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. A random experiment is 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. 8 Important w The actual outcome of a random experiment cannot be determined in advance. w We can only talk about the probability that a particular outcome will occur.

9. 9 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. Sample Space 1 3 4 5 6 2 is the complete list of all the possible outcomes of an experiment. The sample space

10. 10 Example Rolling a die with an even number uppermost can be an event. Sample Space 135135 246246 Events The desired outcome or outcomes from the sample space.

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

12. 12 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: Notation Events can be described in written form, provided the label is defined before hand.

13. 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. 14 The probability of an event is defined as the number of members in an event divided by the number of members in the sample space. Probability of an event 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. 15 Example When rolling a die: {The event of an even number} The probability of A is:

16. 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. 17 The complement of an event are those members of the sample space that are not contained within the event. Complement of an event Complement rule

18. 18 Example When rolling a die, event A is defined as the even numbers. Find : Sample Space A 246246 135135

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

20. 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. 21 Sample Space Find the intersection of A and B. 135135 246246 It’s just me, all alone…

22. 22 Find the probability of A and B.

23. 23 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... Put your hand up if you look like me!

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

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

26. 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. 27 Sample Space Find the probability of X or Y. 135135 246246 All except for me !!

28. 28 Therefore the probability of X or Y is

29. 29 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: Addition formula

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

31. 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. 32 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.

33. 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