Probability Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Probability

Probability The calculated likelihood that a given event will occur Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics The calculated likelihood that a given event will occur

Methods of Determining Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Empirical Experimental observation Example – Process control Theoretical Uses known elements Example – Coin toss, die rolling Subjective Assumptions Example – I think that . . . In practice, engineers will often blend these approaches. For instance, engineers will assume that each one of the widgets produced at a factory has the same (unknown) chance of failure, then make observations to determine the likelihood of failure.

Probability Components Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Experiment An activity with observable results Sample Space A set of all possible outcomes Event A subset of a sample space Outcome / Sample Point The result of an experiment For example, I might test a brass sample to find its tensile strength. [experiment] It might break under any load from 0 to 1000 pounds in my tester, or it may not break at all. [sample space] One possible outcome is that a sample could break at 200 pounds. [Event] When I perform the test, it breaks at a particular load. [Outcome]

Probability What is the probability of a tossed coin landing heads up? Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics What is the probability of a tossed coin landing heads up? Experiment Sample Space Probability Tree Event The probability tree lists all possible outcomes. Many experiments have far too many possible results to write out a probability tree. Outcome

Probability Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics A way of communicating the belief that an event will occur. Expressed as a number between 0 and 1 fraction, percent, decimal, odds Total probability of all possible events totals 1 We’ll deal only with probabilities in this lesson. The odds of an event are slightly different than the probabilities, but if you know one, you can find the other.

Relative Frequency Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics The number of times an event will occur divided by the number of opportunities = Relative frequency of outcome x = Number of events with outcome x n = Total number of events Expressed as a number between 0 and 1 fraction, percent, decimal, odds Total frequency of all possible events totals 1 We’ll deal only with probabilities in this lesson. The odds of an event are slightly different than the probabilities, but if you know one, you can find the other.

Probability What is the probability of a tossed coin landing heads up? Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics What is the probability of a tossed coin landing heads up? How many desirable outcomes? 1 How many possible outcomes? Probability Tree 2 We can calculate the probability of a fair coin flip easily because heads and tails are equally likely. We sometimes call the number of desirable outcomes successes. It is more appropriate to define them as “outcomes”. For instance, if we want to know the probability of catching a cold, we’d put that in the numerator – but a cold is hardly desirable! What is the probability of the coin landing tails up?

Probability Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics What is the probability of tossing a coin twice and it landing heads up both times? HH How many desirable outcomes? 1 HT How many possible outcomes? 4 Teacher note: The most common mistake in this sort of calculation occurs when kids work too fast. There are three possibilities: 2 heads, 2 tails, or 1 of each. This would provide an answer of 1/3 instead of ¼, because there are really 4 possibilities – HT and TH are different outcomes. TH TT

Probability Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics 3rd HHH What is the probability of tossing a coin three times and it landing heads up exactly two times? 2nd HHT 1st How many desirable outcomes? HTH 3 HTT How many possible outcomes? THH 8 THT TTH TTT

Binomial Process Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Each trial has only two possible outcomes yes-no, on-off, right-wrong Trial outcomes are independent Tossing a coin does not affect future tosses **The probability of heads occurring on subsequent coin flips does not change. Notice that a binomial does not have to be a 50-50 chance. Getting a “6” on a die roll is also binomial (you either get the 6 or you don’t).

Bernoulli Process P = Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics P = Probability x = Number of times for a specific outcome within n trials n = Number of trials p = Probability of success on a single trial q = Probability of failure on a single trial ! = factorial – product of all integers less than or equal *Technically a Bernoulli process happens only once (flip one coin), while a binomial process comes by adding many Bernoulli processes. The formula here is for a binomial process (combining the results of n independent Bernoulli trials). n! (read “n factorial”) is the product of all integers from 1 to n. For instance, 5! = 5 x 4 x 3 x 2 x 1 = 120 nCx (read “n choose x”) is the number of distinct groups of x things that can be chosen from n distinct things. Most scientific and graphing calculators have a ! key and a nCx key. Both functions can often be found in a probability menu.

Probability Distribution Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics What is the probability of tossing a coin three times and it landing heads up two times?

Probability Law of Large Numbers Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics The more trials that are conducted, the closer the results become to the theoretical probability Trial 1: Toss a single coin 5 times H,T,H,H,T P = .600 = 60% Trial 2: Toss a single coin 500 times H,H,H,T,T,H,T,T,……T P = .502 = 50.2% Theoretical Probability = .5 = 50% It is still possible to flip a fair coin many times in a row and get significantly different numbers of heads and tails, but this is very unlikely.

Probability AND (Multiplication) P(A and B) = PA∙PB Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics AND (Multiplication) Independent events occurring simultaneously Product of individual probabilities If events A and B are independent, then the probability of A and B occurring is: P(A and B) = PA∙PB Independence means that one event’s outcome doesn’t affect the other event’s outcome.

Probability AND (Multiplication) 1 6 1 6 Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics AND (Multiplication) What is the probability of rolling a 4 on a single die? 1 How many desirable outcomes? 6 How many possible outcomes? What is the probability of rolling a 1 on a single die? 1 How many desirable outcomes? 6 How many possible outcomes? We can multiply these probabilities because they are independent. What is the probability of rolling a 4 and then a 1 in sequential rolls?

Probability OR (Addition) P(A or B) = PA + PB Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics OR (Addition) Independent events occurring individually Sum of individual probabilities If events A and B are mutually exclusive, then the probability of A or B occurring is: P(A or B) = PA + PB If A and B are mutually exclusive, it is not possible for both to occur at the same time. For example, you might want to know the probability of drawing either one of the two cards: a 2 of diamonds or a 2 of spades from a single draw.

Probability OR (Addition) 1 6 1 6 Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics OR (Addition) What is the probability of rolling a 4 on a single die? 1 How many desirable outcomes? 6 How many possible outcomes? What is the probability of rolling a 1 on a single die? 1 How many desirable outcomes? 6 How many possible outcomes? We can add these probabilities because they’re mutually exclusive. Teacher’s note: if two events aren’t mutually exclusive, we have to add their probabilities and then subtract the probability that they both occur. What is the probability of rolling a 4 or a 1 on a single die?

Probability NOT P = 1 - P(A) Independent event not occurring Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics NOT Independent event not occurring 1 minus the probability of occurrence P = 1 - P(A) What is the probability of not rolling a 1 on a die?

Probability Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Two cards are dealt from a shuffled deck. What is the probability that the first card is an ace and the second card is a face card or a ten? How many cards are in a deck? 52 How many aces are in a deck? 4 Dice rolls and coin flips are classic examples of independent events. Drawing cards is a classic example of a set of dependent events. How many face cards are in deck? 12 How many tens are in a deck? 4

Probability What is the probability that the first card is an ace? Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics What is the probability that the first card is an ace? Since the first card was NOT a face, what is the probability that the second card is a face card? Since the first card was NOT a ten, what is the probability that the second card is a ten?

Probability Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Two cards are dealt from a shuffled deck. What is the probability that the first card is an ace and the second card is a face card or a ten? If the first card is an ace, what is the probability that the second card is a face card or a ten? 31.37%

Conditional Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics P(E|A) = Probability of event E, given A Example: One card is drawn from a shuffled deck. The probability it is a queen is P(queen) = However, if I already know it is face card P(queen | face)=

Conditional Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Probability of two events A and B both occurring = P(A and B) = P(A|B)  P(B) = P(B|A)  P(A) If A and B are independent, then P(A and B) = P(A)  P(B)

Probability Bayes’ Theorem Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Calculates a conditional probability, based on all the ways the condition might have occurred. P( A | E ) = probability of A, given we already know the condition E = **Bayes’ theorem is intuitive to most people, but computing it can be very difficult. P(A1 | E) is read “The probability of A1 given E”: that means it’s the likelihood of event A given what we already know about event E.

Bayes’ Theorem Example Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics LCD screen components for a large cell phone manufacturing company are outsourced to three different vendors. Vendor A, B, and C supply 60%, 30%, and 10% of the required LCD screen components. Quality control experts have determined that .7% of vendor A, 1.4% of vendor B, and 1.9% of vendor C components are defective. If a cell phone was chosen at random and the LCD screen was determined to be defective, what is the probability that the LCD screen was produced by vendor A? Note that the vendors are mutually exclusive: any given screen is outsourced to only one company.

Bayes’ Theorem Example Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Bayes’ Theorem Example Notation Used: P = Probability D = Defective A, B, and C denote vendors Unknown to be calculated: Probability the screen is from A, given that it is defective

Bayes’ Theorem Example Known probabilities: Probability the screen is from A Probability the screen is from B Probability the screen is from C

Bayes’ Theorem Example Known conditional probabilities: Probability the screen is defective given it is from A Probability the screen is defective given it is from B Probability the screen is defective given it is from C

Bayes’ Theorem Example: Defective Part Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics Bayes’ Theorem Example: Defective Part = P(screen is defective AND from A) P(screen is defective from anywhere)

LCD Screen Example Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics

LCD Screen Example Probability Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statistics If a cell phone was chosen at random and the LCD screen was determined to be defective, what is the probability that the LCD screen was produced by vendor B? If a cell phone was chosen at random and the LCD screen was determined to be defective, what is the probability that the LCD screen was produced by vendor C? This is calculated in the same way as for vendor A. The denominator (.0103) will be the same in all three cases. P (produced by B| defective) (probability the phone was produced by vendor B given the phone was defective) = .30*.014/.0103 = .4078 P (produced by C| defective) (probability the phone was produced by vendor C given the phone was defective) = .1*.019/.0103 = .1845 Notice that adding all three probabilities gives us .4078 + .4078 + .1845 = 1 (if we ignore rounding error) because the probability that the phone was manufactured by someone is 100%. Notice also that even though vendor A is the best (only 0.7% defective parts), it is tied for the most defective phones because it manufactures so many phones.