Chapter 3

1. Chapter 3 Probability

2. PROBABILITY AND STATISTICS CHAPTER 3 NOTES SECTION 1 I. REVIEW The basic concepts of probability will provide the foundation for our later study of methods used in statistical inference. A. Basic Terms 1. Statistical Experiment: An action from which an outcome, response, count, or measurement is obtained. a) A game player might roll a die and count the number of dots on top. b) A quality control officer, working for a company that manufactures radios, might select one radio from inventory (at random) and determine whether it is defective or not defective. c) A psychologist might give a Differential Aptitude Test (DAT) and determine a percentile score on the Verbal Reasoning sub test. 2. Sample Space: The set of possible outcomes of a particular experiment a) outcomes for the roll of a die = {1,2,3,4,5,6} b) outcomes for quality control = {defective, not defective} c) outcomes for DAT = {1st percentile, 2nd percentile, 3rd percentile,…..99th percentile}

3. PROBABILITY AND STATISTICS CHAPTER 3 NOTES SECTION 1 3. Event: A subset of the sample space. It is usually denoted with a capital letter. a) Examples of events for the die roll: 1) A = roll is a 4 = {4} 2) B = roll is at least a five = {5, 6} 3) C = roll is a prime number = {2, 3, 5} 4) D = roll is less than 9 = {1, 2, 3, 4, 5, 6} b) Examples of events for radio quality control 1) Q = radio is defective 2) R = radio is not defective c) Examples of events for the DAT 1) S = score is 85th percentile = {85} 2) T = score is at least in the 90th percentile = {90, 91, 92, 93, 94, 95, 96, 97, 98, 99} 4. Relative Likelihood: a) called the Probability of an event (E). b) Denoted P(E) and read as “probability of E”. c) The probability of event B as described earlier as “roll at least a five” can be written as P(at least a five) or P(5 or 6), or simply as P(B).

4. PROBABILITY AND STATISTICS CHAPTER 3 NOTES SECTION 1 B. Methods of determining P(E) 1. Intuition a) I feel like I have a 90% chance of getting the job I applied for. b) Lacks a scientific basis. c) Is not used in statistical studies. 2. Equally probable outcomes a) For a roll of a fair (unloaded) die, outcomes 1,2,3,4,5,6 are equally probable. b) f = number of ways favorable to the event and n = number of outcomes in the sample space. 1) P(roll is a 4) = 1/6 2) P(roll is at least a 5) = 2/6 or 1/3. 3) P(roll is less than 9) = 6/6 or 1. 3. Relative Frequency (Statistical Probability) a) From an inventory of 53 radios, 17 were found to be defective. A radio is selected at random. Find the probability it is defective. 1) f = 17 (the number of defective radios) – In this case, a defective radio is a favorable outcome.

5. PROBABILITY AND STATISTICS CHAPTER 3 NOTES SECTION 1 2) n = 53 (the number of trials in the sample) – The number of radios tested. a. Let Q = event that the radio is not defective. b. P(Q) = 17/53 or .32. C. Range of probability values 1. If an event is certain to happen, it is assigned the probability of 1 = 100%. a) The probability of the event G = the roll of a die will be less than 15 = 1. In symbols: P(G) = 1. 1) This happens since there are 6 favorable ways out of 6 possible outcomes. (All numbers on a die are less than 15). 2) Thus, P(roll is less than 15) = 6/6 = 1. 2. If an event is impossible it is assigned a probability of 0. a) Roll a die and let N = roll is a 9. There are 0 favorable ways out of 6 outcomes. (There are no 9’s on a 6 sided die). b) Thus, P(roll is a 9) = 0/6 = 0. 3. All probability values range between 0 and 1, inclusive. a) Probability values can never be negative. b) Probability values can never be greater than 1.

6. PROBABILITY AND STATISTICS CHAPTER 3 NOTES SECTION 1 D. Complementary Events E´ = not E. 1. The complement of an event E is an event denoted E´. This can be read as “not E” or “complement of E”. 2. The event E´ consists of all the possible outcomes from the sample space which are not in event E. a) If B = at least 5 = 5 or 6 = {5, 6}, then its complement, B´ = less than a 5 = 1, 2, 3, 4 = {1, 2, 3, 4} b) If Q = radio is defective, Q´ = radio is not defective 3. P(E´) = 1 – P(E) a) If P(B) = 1/3, then P(B´) = 1 – 1/3 = 2/3. b) If P(Q) = .32, then P(Q´) = 1 - .32 = .68. 4. P(E´) + P(E) will always = 1. a) The probabilities of complementary events ALWAYS add up to exactly 1.

7. PROBABILITY AND STATISTICS CHAPTER 3 NOTES SECTION 3-1 EXAMPLES You work in a hospital which must be staffed 7 days a week. Each week, you get one day off which is picked at random by your supervisor. Suppose one day of the week will be randomly picked as your day off for next week. A. List the sample space Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday B. List the event A that you will have a day off on the weekend. Saturday, Sunday C. Find the probability that you will have a day off on the weekend. 2/7 = .2857 D. Describe the event “not A”. Do not get a weekend day off E. Find the probability of “not A” 5/7 = .7143 ALSO, P(not A) would equal 1- P(A), as they are complements. So, could have said 1-.2857 = .7143.

8. PROBABILITY AND STATISTICS CHAPTER 3 NOTES SECTION 3-1 EXAMPLES A spokesperson for United Airlines said there is a growing demand among fliers for special meals (vegetarian, low fat, diabetic, etc.). Out of 60,000 meals they serve every day, 6,000 are special. A passenger is selected at random and asked to select a meal. A. List the sample space Special, Regular (not special) B. Let B = Person orders a special meal. Find P(B) P(B) = 6,000/60,000 = .1 C. Find P(not B). P(not B) = 1 – P(B) = 1 - .1 = .9