1. Chapter 4. Probability: The Study of Randomness

2. Uses of Probability Gambling Business Product preferences of consumers
Rate of returns on investments
Engineering
Defective parts
Physical Sciences
Locations of electrons in an atom
Computer Science
Flow of traffic or communications

3. 4.1: Randomness - Goals Be able to state why probability is useful.
Be able to state what randomness and probability mean.
Be able to identify where randomness occurs in particular situations.
Be able to state when trials are independent.

4. Trial and Experiment
Trial
Experiment

5. Random and Probability
We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.
The probability of any outcome of a chance process is the proportion of times the outcome would occur in a very long series of repetitions.

6. Frequentist Interpretation

7. Independent
Independent: the outcome of each situation is not influenced by the result of the previous trial
Example
What is the probability of drawing a heart?
What is the probability that I will draw a heart on the second draw?

8. 4.2: Probability Models - Goals
Be able to write down a sample space in specific circumstances.
Be able to state and apply the five probability rules (this goal will reappear later)
Be able to determine what type of probability is given in a certain situation.
Be able to assign probabilities assuming an equally likelihood assumption.

9. Probability Models
The sample space S of a chance process is the set of all possible outcomes.
An event is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space.
A probability model is a description of some chance process that consists of two parts: a sample space S and a probability for each outcome.

10. Sample Space
What is the sample space in the following situations? Are each of the outcomes equally likely?
I roll one 4-sided die.
I roll two 4-sided dice.
I toss a coin until the first head appears.
A mortgage can be classified as fixed rate (F) or variable (V) and we are considering 2 houses.
The number of minutes that a college student uses their cell phone in a day.

11. Probability Rules
Rule 1. The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1. Rule 2. If S is the sample space in a probability model, then P(S) = 1. Rule 3. addition rule for disjoint events: If A and B are disjoint, P(A or B) = P(A) + P(B). Rule 4: The complement of any event A is the event that A does not occur, written AC. P(AC) = 1 – P(A).

12. Examples: Probability Rules
Additivity:
Roll two 4-sided dice: What is the probability that the sum is 2 or 3?
Mortgage: What is the probability that both houses have the same type of mortgage?

13. Examples: Probability Rules
Compliment:
Roll two 4-sided dice: What is the probability that the sum is greater than 2?
Mortgage: What is the probability that both houses do not have fixed mortgages?

14. Types of Probabilities
Subjective
Empirical
𝑃 𝐴 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑎𝑡 𝐴 𝑜𝑐𝑐𝑢𝑟𝑠 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠
Theoretical (equally likely)
𝑃 𝐴 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝐴 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑆

15. Example: Types of Probabilities
For each of the following, determine the type of probability and then answer the question.
What is the probability of rolling a 2 on a fair 4-sided die?
What is the probability of having a girl in the following community?
What is the probability that Purdue Men’s Basketball team will beat IU later this season?
Girl
0.52
Boy
0.48

16. Examples: Legitimate Probabilities
Which of the following probabilities are legitimate? Why or why not?
Outcome #1 #2 #3 #4 #5 1
0.25
0.1
0.5
1.1 2
-0.2 3
0.3 4
0.4

17. Probability Rules
Rule 5. Multiplication Rule for Independent Events. Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent: P(A and B) = P(A)  P(B)

18. Example: Independence
Are the following events independent or dependent?
Winning at the Hoosier (or any other) lottery.
The marching band is holding a raffle at a football game with two prizes. After the first ticket is pulled out and the winner determined, the ticket is taped to the prize. The next ticket is pulled out to determine the winner of the second prize.

19. Example: Independence
Deal two cards without replacement
A = 1st card is a heart B = 2nd card is a heart
C = 2nd card is a club.
Are A and B independent?
Are A and C independent?
2. Repeat 1) with replacement.

20. Disjoint vs. Independent
In each situation, are the following two events
a) disjoint and/or b) independent?
Draw 1 card from a deck
A = card is a heart B = card is not a heart
Toss 2 coins
A = Coin 1 is a head B = Coin 2 is a head
Roll two 4-sided dice.
A = red die is 2 B = sum of the dice is 3

21. Example: Complex Multiplication Rule (1)
The following circuit is in a series. The current will flow only if all of the lights work. Whether a light works is independent of all of the other lights. If the probability that A will work is 0.8, P(B) = 0.85 and P(C) = 0.95, what is the probability that the current will flow? A B C

22. Example: Complex Multiplication Rule (2)
The following circuit to the right is parallel. The current will flow if at least one of the lights work. Whether a light works is independent of all of the other lights. If the probability that A will work is 0.8, P(B) = 0.85 and P(C) = 0.95, what is the probability that the current will flow? A B C

23. Example: Complex Multiplication Rule (3)
A diagnostic test for a certain disease has a specificity of 95%. The specificity is the same as true negative, that is the test is negative when the person doesn’t have the disease. a) What is the probability that one person has a false positive (the test is positive when they don’t have the disease)? b) What is the probability that there is at least one false positive when 50 people who don’t have the disease are tested?

24. 4.3: Random Variables - Goals
Be able to define what a random variable is.
Describe the probability distribution of a discrete random variable.
Use the distribution of a discrete random variable to calculate probabilities of events.
Describe the probability distribution of a continuous random variable.
Use the distribution of a continuous random variable to calculate probabilities of events.

25. Random Variables
A random variable takes numerical values that describe the outcomes of some chance process.
The probability distribution of a random variable gives its possible values and their probabilities.

26. Discrete Random Variables
A discrete random variable X takes a fixed set of possible values with gaps between.
They are usually displayed in table form
These probabilities must satisfy the following:
0 ≤ pi ≤ 1
Sum of all the pi’s is 1
value x1 x2 …
probability p1 p2

27. Examples: Probability Histograms

28. Example: Discrete Random Variable
In a standard deck of cards, we want to know the probability of drawing a certain number of spades when we draw 3 cards. Let X be the number of spades that we draw.
What is the distribution?
What is the probability that you draw at least 1 spade?
What is the probability that you draw at least 2 spades?

29. Example: Discrete (cont.)

30. Normal Distribution: Example
A particular rash has shown up in an elementary school. It has been determined that the length of time that the rash will last is normally distributed with mean 6 days and standard deviation 1.5 days.
What is the percentage of students that have the rash for longer than 8 days?
What is the percentage of students that the rash will last between 3.7 and 8 days?

31. 4.4: Means and Variances of Random Variables - Goals
Be able to use a probability distribution to find the mean of a discrete (or continuous) random variable.
Be able to use the law of large numbers to describe the behavior of the sample mean.
Calculate means using the rules for means.
Be able to use a probability distribution to find the variance of a discrete (or continuous) random variable.
Calculate variances using the rules for variances for both correlated and uncorrelated random variables.

32. Formulas for the Mean of a Random Variable
Discrete
𝐸 𝑋 = 𝜇 𝑋 = 𝑖 𝑥 𝑖 𝑝 𝑖

33. Example: Expected value
What is the expected value of the following: a) A fair 4-sided die X 1 2 3 4
Probability
0.25

34. Statistical Estimation
What would happen if we took many samples?
Population
Sample ?

35. Law of Large Numbers
Draw independent observations at random from any population with finite mean µ. The law of large numbers says that, as the number of observations drawn increases, the sample mean of the observed values gets closer and closer to the mean µ of the population.
Our intuition doesn’t do a good job of distinguishing random behavior from systematic influences. This is also true when we look at data.

36. Rules for Means
Rule 1: If X is a random variable and a and b are fixed numbers, then: µa+bX = a + bµX Rule 2: If X and Y are random variables, then: µXY = µX  µY Rule 3: If X is a random variable and g is a function of X, then: 𝐸 𝑔 𝑋 = 𝑔( 𝑥 𝑖 ) 𝑝 𝑖

37. Example: Expected Value
An individual who has automobile insurance form a certain company is randomly selected. Let X be the number of moving violations for which the individual was cited during the last 3 years. The distribution of X is a) Verify that E(X) = b) If the cost of insurance depends on the following function of accidents, g(y) = (100y -15), what is the expected value of the cost of the insurance? X 1 2 3 px
0.60
0.25
0.10
0.05

38. Example: Expected Value
Five individuals who have automobile insurance from a certain company are randomly selected. Let X and Y be two different accident profiles in this insurance company: E(X) = 0.60 E(Y) = 0.95 c) What is the expected value the total number of accidents of the people if 2 of them have the distribution in X and 3 have the distribution in Y? X 1 2 3 px
0.60
0.25
0.10
0.05 Y 1 2 3 pY
0.40
0.35
0.15
0.10

39. Example: Expected value
An individual who has automobile insurance form a certain company is randomly selected. Let X be the number of moving violations for which the individual was cited during the last 3 years. The distribution of X is d) Calculate E(X2). X 1 2 3 px
0.60
0.25
0.10
0.05

40. Variance of a Random Variable
𝑠 2 = 𝑖=1 𝑛 𝑥 𝑖 − 𝑥 2 𝑛−1 Var(X)=E X− 𝜇 𝑋 2 = ( 𝑥 𝑖 −  X ) 2 ∙ 𝑝 𝑖 = E(X2) – (E(X))2 𝜎 𝑋 = 𝑉𝑎𝑟(𝑋)

41. Example: Variance
An individual who has automobile insurance form a certain company is randomly selected. Let X be the number of moving violations for which the individual was cited during the last 3 years. The distribution of X is e) Calculate Var(X). X 1 2 3 px
0.60
0.25
0.10
0.05

42. Rules for Variance
Rule 1: If X is a random variable and a and b are fixed numbers, then: σ2a+bX = b2σ2X Rule 2: If X and Y are independent random variables, then: σ2XY = σ2X + σ2Y Rule 3: If X and Y have correlation ρ, then: σ2XY = σ2X + σ2Y  2ρσXσY

43. Correlation

44. Example: Variance
An individual who has automobile insurance form a certain company is randomly selected. Let X be the number of moving violations for which the individual was cited during the last 3 years. The distribution of X is
Calculate the variance of this distribution.
If the cost of insurance depends on the following function of accidents, g(y) = (100y -15), what is the standard deviation of the cost of the insurance? X 1 2 3 px
0.60
0.25
0.10
0.05

45. Example: Variance X 1 2 3 px 0.60 0.25 0.10 0.05 Y 1 2 3 pY 0.40 0.35
5 individuals who have automobile insurance from a certain company are randomly selected. Let X and Y be two different independent accident profiles in this insurance company: Var(X) = 0.74 Var(Y) = 0.95 What is the standard deviation of the difference between the 2 who have insurance using the X distribution and the 3 who have insurance using the Y distribution? X 1 2 3 px
0.60
0.25
0.10
0.05 Y 1 2 3 pY
0.40
0.35
0.15
0.10

46. 4.5: General Probability Rules - Goals
Apply the five rules of probability (again).
Apply the generation addition rule.
Be able to calculate conditional probabilities.
Apply the general multiplication rule.
Be able to use tree diagram.
Use Bayes’s rule to find probabilities.
Determine if two events with positive probability are independent.

47. Probability Rules
Rule 1. The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1. Rule 2. If S is the sample space in a probability model, then P(S) = 1. Rule 3. If A and B are disjoint, P(A or B) = P(A) + P(B). Rule 4: For any event A, P(AC) = 1 – P(A). Rule 5: If A and B are independent: P(A and B) = P(A) P(B)

48. General Addition Rule
P(A or B) = P(A) + P(B) – P(A and B) Select a card at random from a deck of cards. What is the probability that the card is either an Ace or a Heart?

49. Example: Venn Diagrams
At a certain University, the probability that a student is a math major is 0.25 and the probability that a student is a computer science major is In addition, the probability that a student is a math major and a student science major is a) What is the probability that a student is a math major or a computer science major? b) What is the probability that a student is a computer science major but is NOT a math major?

50. Conditional Probability
423/what-is-your-favorite-data-analysis-cartoon

51. Conditional Probability: Example
A news magazine publishes three columns entitled "Art" (A), "Books" (B) and "Cinema" (C). Reading habits of a randomly selected reader with respect to these columns are a) What is the probability that a reader reads the Art column given that they also read the Books column? b) What is the probability that a reader reads the Books column given that they also read the Art column?
Read Regularly A B C
A and B
A and C
B and C
Probability
0.14
0.23
0.37
0.08
0.09
0.13

52. Example: General Multiplication Rule
Suppose that 8 good and 2 defective fuses have been mixed up. To find the defective fuses we need to test them one-by-one, at random. Once we test a fuse, we set it aside. a) What is the probability that we find both of the defective fuses in the first two tests? b) What is the probability that when testing 3 of the fuses, the first tested fuse is good and the last two tested are defective?

53. Independence Revisited
General multiplication rule: P(A and B) = P(A) P(B|A) If A and B are independent: P(A and B) = P(A)  P(B) Therefore, if A and B are independent: P(B|A) = P(B)

54. Example: Tree Diagram/Bayes’s Rule
A diagnostic test for a certain disease has a 99% sensitivity and a 95% specificity. Only 1% of the population has the disease in question. If the diagnostic test reports that a person chosen at random from the population tests positive, what is the probability that the person does, in fact, have the disease?

55. Bayes’s Rule
Suppose that a sample space is decomposed into k disjoint events A1, A2, … , Ak —none of which has a 0 probability—such that
𝑖=1 𝑘 𝑃( 𝐴 𝑖 ) =1
Let B be any other event such that P(B) is not 0. Then
𝑃 𝐴 𝑗 𝐵 = 𝑃 𝐵 𝐴 𝑗 𝑃( 𝐴 𝑗 ) 𝑖=1 𝑘 𝑃(𝐵| 𝐴 𝑖 )𝑃( 𝐴 𝑖 )

56. Law of Total Probability5 4 1 6 3
B and 4
B and 6
B and 3
B and 7 2 7 B