Probability A Brief Look

A Few Terms Probability represents a standardized measure of chance, and quantifies uncertainty. Let S = sample space which is the set of all possible outcomes. An event is a set of possible outcomes that is of interest. If A is an event outcome, then P(A) is the probability that event outcome A occurs.

Sample Space, A’s and P(A)’s What is the chance that it will rain today? The number of maintenance calls for an old photocopier is twice that for the new photocopier. What is the chance that the next call will be regarding an old photocopier? If I pull a card out of a pack of 52 cards, what is the chance it’s a spade?

Union and Intersection of Events The intersection of events A and B refers to the probability that both event A and event B occur. The union of events A and B refers to the probability that event A occurs or event B occurs or both events, A & B, occur. and either/or

Mutually Exclusive Events Mutually exclusive events can not occur at the same time. S S Mutually Exclusive Events Not Mutually Exclusive Events

Roommate profile Distribution Frequency - Counts Snores Doesn’t Snore Parties 150 100 250 Doesn’t Party 200 550 750 350 650 1000 Relative Frequency - Probability Snores Doesn’t Snore Parties 0.15 0.10 0.25 Doesn’t Party 0.20 0.55 0.75 0.35 0.65 1.00 University of South Carolina; Slide 6

What is the probability that a randomly chosen roommate will snore? Snores Doesn’t Snore Parties 150 100 250 Doesn’t Party 200 550 750 350 650 1000

What is the probability that a randomly chosen roommate will like to party? Snores Doesn’t Snore Parties 150 100 250 Doesn’t Party 200 550 750 350 650 1000

What is the probability that a randomly chosen roommate will snore or like to party? Snores Doesn’t Snore Parties 150 100 250 Doesn’t Party 200 550 750 350 650 1000

The Union of Two Events If events A & B intersect, you have to subtract out the “double count”. If events A & B do not intersect (are mutually exclusive), there is no “double count”.

Given that a randomly chosen roommate snores, what is the probability that he/she likes to party? Doesn’t Snore Parties 150 100 250 Doesn’t Party 200 550 750 350 650 1000 University of South Carolina; Slide 11

Conditional Probability The conditional probability of B, given that A has occurred: Given University of South Carolina; Slide 12

Probability of Intersection Solving the conditional probability formula for the probability of the intersection of A and B:

When , we say that Events B and A are Independent. The basic idea underlying independence is that information about event A provides no new information about event B. So “given event A has occurred”, doesn’t change our knowledge about the probability of event B occurring.

There are 10 light bulbs in a bag, 2 are burned out. If we randomly choose one and test it, what is the probability that it is burned out? If we set that bulb aside and randomly choose a second bulb, what is the probability that the second bulb is burned out?

Near Independence EX: Car company ABC manufactured 2,000,000 cars in 2009; 1,500,000 of the cars had anti-lock brakes. If we randomly choose 1 car, what is the probability that it will have anti-lock brakes? If we randomly choose another car, not returning the first, what is the probability that it will have anti-lock brakes?

Independence Sampling with replacement makes individual selections independent from one another. Sampling without replacement from a very large population makes individual selection almost independent from one another

Probability of Intersection Probability that both events A and B occur: If A and B are independent, then the probability that both occur:

Test for Independence If , then A and B are independent events. If A and B are not independent events, they are said to be dependent events. University of South Carolina; Slide 19

Are snoring or not and partying or not independent of one another? Snores Doesn’t Snore Parties 150 100 250 Doesn’t Party 200 550 750 350 650 1000 University of South Carolina; Slide 20

Arrange the counts so that snoring and partying are independent of one another. Snores Doesn’t Snore Parties Doesn’t Party 1000 University of South Carolina; Slide 21

Complementary Events The complement of an event is every outcome not included in the event, but still part of the sample space. The complement of event A is denoted A. Event A is not event A. The complement of an event is every outcome not included in the event, but still part of the sample space. The complement of event A is denoted A. Event A is not event A. S: A A University of South Carolina; Slide 22

All mutually exclusive events are complementary. ????????????????? All mutually exclusive events are complementary. True False University of South Carolina; Slide 23

Probability Rules 0 < P(A) < 1 Sum of all possible mutually exclusive outcomes is 1. Probability of A or B: Probability of A or B when A, B are mutually exclusive: University of South Carolina; Slide 24

Probability Rules Continued Probability of B given A: Probability of A and B: Probability of A and B when A, B are independent: University of South Carolina; Slide 25

Probability Rules Continued If A and A are compliments: or University of South Carolina; Slide 26

So Let’s Apply the Rules We purchase 30% of our parts from Vendor A. Vendor A’s defective rate is 5%. What is the probability that a randomly chosen part is defective and from Vendor A? University of South Carolina; Slide 27

???????????? We are manufacturing widgets. 50% are red, 30% are white and 20% are blue. What is the probability that a randomly chosen widget will not be white? University of South Carolina; Slide 28

???????????? When a computer goes down, there is a 75% chance that it is due to overload and a 15% chance that it is due to a software problem. There is an 85% chance that it is due to an overload or a software problem. What is the probability that both of these problems are at fault? University of South Carolina; Slide 29

???????????? It has been found that 80% of all accidents at foundries involve human error and 40% involve equipment malfunction. 35% involve both problems. If an accident involves an equipment malfunction, what is the probability that there was also human error? University of South Carolina; Slide 30

????????????????? Four electrical components are connected in series. The reliability (probability the component operates) of each component is 0.90. If the components are independent of one another, what is the probability that the circuit works when the switch is thrown? A B C D University of South Carolina; Slide 31

An automobile manufacturer gives a 5-year/75,000-mile warranty on its drive train. Historically, 7% of the manufacturer’s automobiles have required service under this warranty. Consider a random sample of 15 cars. If we assume the cars are independent of one another, what is the probability that no cars in the sample require service under the warranty? What is the probability that at least one car in the sample requires service? University of South Carolina; Slide 32

Consider the following electrical circuit: B C 0.95 0.95 0.95 The probability on the components is their reliability (probability that they will operate when the switch is thrown). Components are independent of one another. What is the probability that the circuit will not operate when the switch is thrown? University of South Carolina; Slide 33

Consider the electrical circuit below Consider the electrical circuit below. Probabilities on the components are reliabilities and all components are independent. What is the probability that the circuit will work when the switch is thrown? A 0.90 C 0.95 B 0.90 University of South Carolina; Slide 34

The number of maintenance calls for an old photocopier is twice that for the new photocopier. Outcomes Old Machine New Machine Probability 0.67 0.33 Which of the following series of events would most cause you to question the validity of the above probability model? Two maintenance calls for an old machine followed by a call for a new machine. Two maintenance calls for new machines followed by a call for an old machine. Three maintenance calls in a row for an old machine. Three maintenance calls in a row for a new machine University of South Carolina; Slide 35

????? What is the probability that at least 2 people in this class (n=39) have the same birthday – Month and day? Year has 365 days – forget leap year. Equally likelihood for each day University of South Carolina; Slide 36

Multiplication of Choices If an operation can be performed in n1 ways, and if for each of these a second operation can be performed in n2 ways, and for each of the first two a third operation can be performed in n3 ways, and so forth, then the sequence of k operations can be performed in n1 · n2 ·…· nk ways. University of South Carolina; Slide 37

Multiplication of Choices There are 5 processes needed to manufacture the side panel for a car: clean, press, cut, paint, polish. Our plant has 6 cleaning stations, 3 pressing stations, 8 cutting stations, 5 painting stations, and 8 polishing stations. How many different “pathways” through the manufacturing exist? What is the number of “pathways” that include a particular pressing station? What is the probability that a panel follows any particular path? What is the probability that a panel goes through pressing station 1?

Classical Definition of Probability If an experiment can result in any one of N different, but equally likely, outcomes, and if exactly n of these outcomes corresponds to event A, then the probability of event A is University of South Carolina; Slide 39

Counting Suppose there are 3 vendors and we want to choose 2. How many possible combinations of 2 can be chosen from the 3 vendors? University of South Carolina; Slide 40

Counting - Combinations The number of combinations of n distinct objects taken r at a time is “n choose r” University of South Carolina; Slide 41

Factorial Reminder n! = n · (n-1) · (n-2) ·….. 1! = 1 0! = 1 EX: 4! = (4)(3)(2)(1) = 24 1! = 1 0! = 1 University of South Carolina; Slide 42

Counting 9 out of 100 computer chips are defective. We choose a random sample of n=3. How many different samples of 3 are possible? How many of the samples of 3 contain exactly 1 defective chip? What is the probability of choosing exactly 1 defective chip in a random sample of 3? What is the probability of choosing at least 1 defective chip in a random sample of 3? University of South Carolina; Slide 43

What is the probability that there will be at least 1 woman chosen? This class consists of 5 women and 34 men. If we randomly choose 4 people, what is the probability that there will be no women chosen? What is the probability that there will be at least 1 woman chosen? University of South Carolina; Slide 44