Probability Class 29 1

Homework Check Assignment: Chapter 7 – Exercise 7.5, 7.7, 7.10 and 7.12 Reading: Chapter 7 – p

Suggested Answer 3

4

5

6

7 Estimating Probabilities from Observed Categorical Data Assuming data are representative, the probability of a particular outcome is estimated to be the relative frequency (proportion) with which that outcome was observed. Approximate margin of error for the estimated probability is

8 Example 7.5 Night-lights and Myopia Revisited Assuming these data are representative of a larger population, what is the approximate probability that someone from that population who sleeps with a nightlight in early childhood will develop some degree of myopia? Note: = 79 of the 232 nightlight users developed some degree of myopia. Estimated probability is 79/232 = Estimate based on sample of 232 with a margin of error of ~0.066.

9 The Personal Probability Interpretation Personal probability of an event = the degree to which a given individual believes the event will happen. Sometimes subjective probability used because the degree of belief may be different for each individual. For example: the hiring of a particular person Restrictions on personal probabilities: Must fall between 0 and 1 (or between 0 and 100%). Must be coherent.

Probability Definitions and Relationships Sample space: the collection of unique, nonoverlapping possible outcomes of a random circumstance. Simple event: one outcome in the sample space; a possible outcome of a random circumstance. Compound event: an event that includes two or more simple events. Event: a collection of one or more simple events in the sample space; often written as A, B, C, and so on. The probability of an event – P (A)

11 Example 7.6 Days per Week of Drinking Random sample of college students. Q: How many days do you drink alcohol in a typical week? Simple Events in the Sample Space are: 0 days, 1 day, 2 days, …, 7 days Event “4 or more” is comprised of the simple events {4 days, 5 days, 6 days, 7 days}

12 Assigning Probabilities to Simple Events P(A) = probability of the event A Two Conditions for Valid Probabilities 1.Each probability is between 0 and 1. 2.The sum of the probabilities over all possible simple events is 1. Probabilities for Equally Likely Simple Events If there are k simple events in the sample space and they are all equally likely, then the probability of the occurrence of each one is 1/k.

13 Example 7.7 Lottery Event Probabilities Random Circumstance: A three-digit winning lottery number is selected. What is the sample space? Sample Space: {000,001,002,003,...,997,998,999}. There are 1000 simple events. Probabilities for Simple Event: What is the probability of a winning three-digit winning number? ( A specific simple event) Probability any specific three-digit number is a winner is 1/1000. (Assume all three-digit numbers are equally likely)

14 Example 7.7 Lottery Event Probabilities What is the probability of an Event A in which the last digit is a 9? Event A = last digit is a 9 = {009,019,...,999}. Since one out of ten numbers in set, P(A) = 1/10. What is the probability of an Event A in which the last three digits are the same? Event B = three digits are all the same = {000, 111, 222, 333, 444, 555, 666, 777, 888, 999}. Since event B contains 10 events, P(B) = 10/1000 = 1/100.

15 Complementary Events Note: P(A) + P( A C ) = 1 One event is the complement of another event if the two events do not contain any of the same simple events and together they cover the entire sample space. Notation: A C represents the complement of A. Example 7.8 Probability of Not Winning the Lottery A = player buying single ticket wins A C = player does not win P(A) = 1/1000 so P(A C ) = 999/1000

16 Mutually Exclusive Events Two events are mutually exclusive, or equivalently disjoint, if they do not contain any of the same simple events (outcomes). Example 7.9 Mutually Exclusive Events for Lottery Numbers A = all three digits are the same. B = the first and last digits are different The events A and B are mutually exclusive (disjoint), but they are not complementary.

17 Independent and Dependent Events Two events are independent of each other if knowing that one will occur (or has occurred) does not change the probability that the other occurs. Two events are dependent if knowing that one will occur (or has occurred) changes the probability that the other occurs. The definitions can apply either … to events within the same random circumstance or to events from two separate random circumstances.

18 Example 7.10 Winning a Free Lunch Customers put business card in restaurant glass bowl. Drawing held once a week for free lunch. You and Vanessa put a card in two consecutive weeks. Event A = You win in week 1. Event B = Vanessa wins in week 1. Event C = Vanessa wins in week 2. Events A and B refer to the same random circumstance and are not independent. Events A and C refer to different random circumstances and are independent.

19 Example 7.11 Alicia Answering Event A = Alicia is selected to answer Question 1. Event B = Alicia is selected to answer Question 2. P(A) = 1/50. If event A occurs, her name is no longer in the bag, so P(B) = 0. If event A does not occur, there are 49 names in the bag (including Alicia’s name), so P(B) = 1/49. Events A and B refer to different random circumstances, but are A and B independent events? Knowing whether A occurred changes P(B). Thus, the events A and B are not independent.

Homework Assignment: Chapter 7 – Exercise 7.20, 7.29, 7.47 and 7.48 Reading: Chapter 7 – p