1. Probability Probability Principles of EngineeringTM
Unit 4 – Lesson Statistics
Probability

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

3. Methods of Determining Probability
Principles of EngineeringTM
Unit 4 – Lesson 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 that likelihood.

4. Probability Components
Principles of EngineeringTM
Unit 4 – Lesson 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 not break at all. [sample space]
Each test sample will either break at a particular load, or not break at all. [Event]
When I perform the test, I record the result. [Outcome]

5. Probability What is the probability of a tossed coin landing heads up?
Principles of EngineeringTM
Unit 4 – Lesson 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. In many experiments there are far too many possible results to write out a probability tree.
Outcome

6. Probability
Probability
Principles of EngineeringTM
Unit 4 – Lesson Statistics
The number of times an event will occur divided by the number of opportunities
Px = Probability of outcome x
Fx = Frequency of outcome x
Fa = Absolute frequency of all events
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.

7. Probability What is the probability of a tossed coin landing heads up?
Principles of EngineeringTM
Unit 4 – Lesson 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 desirable outcomes “successes”. It’s more exact to say these “Outcomes we’re studying.” For instance, if we want to know the probability of getting 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?

8. Probability
Probability
Principles of EngineeringTM
Unit 4 – Lesson 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 comes when kids go too fast: there are three possibilities: 2 heads, 2 tails, or 1 of each. This would get an answer of 1/3 instead of ¼, because there are really 4 possibilities- HT and TH are different outcomes. TH TT

9. Probability
Probability
Principles of EngineeringTM
Unit 4 – Lesson 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

10. Binomial Process
Probability
Principles of EngineeringTM
Unit 4 – Lesson 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
**Also, the probability of heads on each coin flip doesn’t change.
Notice it doesn’t have to be a chance. Getting a “6” on a die roll is also binomial (you either get the 6 or you don’t).

11. Bernoulli Process P = Probability
Principles of EngineeringTM
Unit 4 – Lesson Statistics
P = Probability
x = Number of times an outcome occurs within n trials
n = Number of trials
p = Probability of success on a single trial
q = Probability of failure on a single trial
*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! (called “n factorial) is the number n times each number smaller than it. For instance, 5! = 5*4*3*2*1 = 120
Most scientific and graphing calculators have a ! key.

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

13. Probability
Law of Large Numbers
Principles of EngineeringTM
Unit 4 – Lesson 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’s still possible to flip a fair coin many times in a row and get significantly different numbers of heads and tails, it’s just very unlikely.

14. Probability AND (Multiplication) P = P(A) x P(B)
Principles of EngineeringTM
Unit 4 – Lesson 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 = P(A) x P(B)
Independence means that one event’s outcome doesn’t effect the other.

15. Probability AND (Multiplication) 1 6 1 6
Principles of EngineeringTM
Unit 4 – Lesson 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’re independent.
What is the probability of rolling a 4 and then a 1 using two dice?

16. Probability OR (Addition) P = P(A) + P(B)
Principles of EngineeringTM
Unit 4 – Lesson 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 = P(A) + P(B)
If A and B are mutually exclusive, they can’t both happen. For instance, you can’t get both “heads” and “tails” on the same coin flip.

17. Probability OR (Addition) 1 6 1 6
Principles of EngineeringTM
Unit 4 – Lesson 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?

18. Probability NOT P = 1 - P(A) Independent event not occurring
Principles of EngineeringTM
Unit 4 – Lesson 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?

19. Probability
Probability
Principles of EngineeringTM
Unit 4 – Lesson 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

20. Probability What is the probability that the first card is an ace?
Principles of EngineeringTM
Unit 4 – Lesson 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?

21. Probability
Probability
Principles of EngineeringTM
Unit 4 – Lesson 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%

22. How about something more complex?
Probability
Principles of EngineeringTM
Unit 4 – Lesson 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.

23. Probability
Bayes’ Theorem
Principles of EngineeringTM
Unit 4 – Lesson Statistics
The probability of an event occurring based upon other event probabilities
**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.

24. LCD Screen Example P = Probability D = Defective
Principles of EngineeringTM
Unit 4 – Lesson Statistics
P = Probability
D = Defective
A, B, and C denote vendors
So how do you find: P A+D = ?
PA x PDA
(PA x PDA) + (PB x PDB) + (PC x PDC)

25. LCD Screen Example Probability Principles of EngineeringTM
Unit 4 – Lesson Statistics

26. LCD Screen Example
Probability
Principles of EngineeringTM
Unit 4 – Lesson 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 = 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’s tied for the most defective phones because it makes so many of them. Manufacturer C has a