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STAT 311: Introductory Probability

http://mrburkemath.blogspot.com/2010/12/snow-days.html

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Part 1: Randomness

http://mikeess-trip.blogspot.com/2011/06/gambling.html

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Applications of Probability

• Games of Chance• Diagnosis of Diseases• Engineering• Biology/agriculture• Physics/Chemistry• Business Management• Computer Science

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Chapter 1: Outcomes, Events, and Sample Spaces

http://www.thescientificcartoonist.com/?p=102

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Sample Spaces: ExamplesFor each of the following examples, determine the

sample set and state the possible values for the elements of the sample set.

1. Tossing Coins: We toss a coin 3 times2. Rolling two 4-sided dies3. Lifetime of a light bulb4. Genetics: Dominant (A=black hair) or recessive

(a = red hair)

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Events: Examples1. Tossing Coins: 3 times

a) Determine the event that there is only one Head.b) Identify in words the event: {HHH,TTT}

2. Rolling two 4-sided diesa) Determine the event that the sum of the two dice is 9.b) Determine the event that the difference between the

numbers of the white and red dies is 1.c) Identify in words the event: {(x, x+1): x {1,2,3} }

3. Lifetime of a light bulba) Determine the event that the light bulb lasts between

100 and 110 hours.b) Identify in words the event: {x | x < 200}

4. Genetics: Dominant (A=black hair) or recessive (a = red hair)a) Determine the event that the hair color is black.b) Identify in words the event: {aa, aA, Aa}

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Example 1.17: Pick ten songsA student hears 10 songs (in a random shuffle mode) on her music player, noting how many of these songs belong to her favorite type of music.a) What is the sample space?b) How many different events are there?c) What is the event, A, that none of the first three songs are

her favorite type of music? How many outcomes are there?d) What is the event, B, that the even-numbered songs are from

her favorite type of music? How many outcomes are there?e) What is the event of A ∩ B? How many outcomes are there?f) What is the event, C, that the last 5 songs are from her

favorite type of music? How many outcomes are there?g) What is the event B ∩ C? How many outcomes are there?

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DeMorgan’s Laws

• Theorem 1.22 DeMorgan’s first law For a finite or infinite collection of events, A1, A2, …

• Theorem 1.23 DeMorgan’s second law For a finite or infinite collection of events, A1, A2, …

c

cj j

j j

A A

c

cj j

j j

A A

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Set Theory: Other Laws• Distributive Laws

Let A, B, and C be subsets of S. Thena) A ∩ (B U C) = (A ∩ B) U (A ∩ C)b) A U (B ∩ C) = (A U B) ∩ (A U C)

• Associative and Commutative LawsLet A, B, and C be subsets of S, Thena) A ∩ B = B ∩ Ab) A U B = B U Ac) A ∩ (B ∩ C) = (A ∩ B) ∩ Cd) A U (B U C) = (A U B) U C

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Example: Subsets, etc.Write down the set of events for each of the

following:Rolling 2 4-sided dice:

A: the red die is a 4 B: the white die is a 2C: The sum of the two dice is 4D: the red die is a 3

a) AC f) Ac\Cb) A ∩ C g) D U (B ∩ C)c) A U Ch) (D U B) ∩ (D U C)d) AC ∩ Ce) AC U C

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Chapter 2: Probability

http://www.cartoonstock.com/directory/p/probability.asp

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Frequentist Interpretation

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

Trial 1Trial 2Trial 30.25

Number of draws

Prop

ortio

n of

Hea

rts

0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

Trial 1Trial 20.25

Number of draws

Prop

ortio

n of

Hea

rts

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Disjoint

• Event A and event B are disjoint.• Both events A and B cannot occur.• Event A and event B have no common

outcomes.

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Axioms 2.4

1. NonnegativityFor each event A, 0 ≤ P(A) ≤ 1

2. CertaintyFor the sample space, S, P(S) = 1

3. AdditivityIf A1, … is a collection of disjoint events, then

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Theorems

Th. 2.5: The probability of the empty set, , is always 0.

Th. 2.6: If A1, A2, …, An is a collection of finitely many disjoint events, then

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Example: Legitimate Probabilities of Sample Spaces

Row # Outcome #1 #2 #3 #4 #5 #61 11 0.0625 0 0.1 0.1 0.05 0.05

2 12 0.0625 0 0.1 0.1 0.05 0.05

3 13 0.0625 0 0.1 0.1 0.05 0.05

4 14 0.0625 0 0.1 0.1 0.05 0.05

5 21 0.0625 0.25 0.1 0.2 0.1 0.1

6 22 0.0625 0.25 0.1 0.2 0.1 0.1

7 23 0.0625 0.25 0.1 0.2 0.1 0.1

8 24 0.0625 0.25 0.1 0.2 0.1 0.1

9 31 0.0625 0 0.1 -0.1 0 0.01

10 32 0.0625 0 0.1 -0.1 0 0.01

11 33 0.0625 0 0.1 -0.1 0 0.01

12 34 0.0625 0 0.1 -0.1 0 0.01

13 41 0.0625 0 0.1 0.05 0.1 0.05

14 42 0.0625 0 0.1 0.05 0.1 0.05

15 43 0.0625 0 0.1 0.05 0.1 0.05

16 44 0.0625 0 0.1 0.05 0.1 0.05

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Section 2.3: Theorems

• Complementation Rule: Th. 2.19: P(Ac) = 1 – P(A)

• Domination Principle:Th. 2.20: If A B, the P(A) ≤ P(B).

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Inclusion/Exclusion(General Addition Rule)

Th. 2.22: For any two events A and B,P(A U B) = P(A) + P(B) – P(A ∩ B)

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Example 2.26

Consider a student who draws cards from a deck. After he draws the card, he replaces the card and then reshuffles the deck. He stops if he draws the ace of spaces.

What is the probability of Bk, when the ace of spaces is found for the first time on the kth draw?

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Example 2.27A student hears ten songs (in a random shuffle mode) on her music player, paying special attention to how many of these songs belong to her favorite type of music. We assume the songs are picked independently of each other and that each song has probability p of being a song of the student’s favorite type.

What is P(Aj) where Aj is the event that exactly j of the 10 songs are from her favorite type of music?

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Chapter 3: Independent Events

http://www.cartoonstock.com/directory/p/probability.asp

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Example: Independence

Roll a red 4 sided die and a white 4 sided die.Let A: event that the red die is a 1

B: event that the white die is a 1 C: event that the sum of the two dice is 4

a) Are events A and B independent?b) Are events A and C independent?

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Example: Disjoint and Independent

Roll a red 4 sided die and a white 4 sided die. Are each of the following disjoint and/or independent?

1) A: event that the red die is a 1 B: event that the red die is a 2

2) A: event that the red die is a 1 B: event that the white die is a 2

3) A: event that the red die is a 1 B: event that the sum of the two dice is 4

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Example: Pairwise Independence

Roll a red 4 sided die and a white 4 sided die.Let A: event that the red die is even

B: event that the white die is even C: event that the sum of the two dice is even

1) Show that A, B, and C are pairwise independent.

2) Show that A ∩ B and C are NOT independent.

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Example: Mutual Independence

Roll a red 6 sided die and a white 6 sided die.Let D: event that the red die is 1 or 2 or 3

E: event that the white die 4 or 5 or 6 F: event that the sum of the two dice is 5

Show that P(D ∩ E ∩ F) = P(D)P(E)P(F) but D, E and F are NOT (mutually) independent events.

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Example 3.19: Independence

A student flips a coin until the tenth head appears. Let A denote the event that at least 3 flips are needed between the 7th and 8th heads; let B denote the event that at least 3 flips are needed between the 8th and 9th heads.1) What would be considered the trial?2) Are A and B independent?

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Example: Independence (cont.)

If the probability that a fuse is good in a particular batch of fuses is 0.8 and each fuse is independent of the other fuses, what is the probability that 2 fuses are bad?

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Theorem 3.24: Good before Bad

Consider a sequence of independent trials, each of which can be classified as good, bad, or neutral, which happen (on any given trial) with probabilities p, q, and 1 – p – q. (We do not necessarily have q = 1 – p here, although that is allowed.) Then the probability that something good happens before something bad happens is

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Chapter 4: Conditional Probability

http://imgs.xkcd.com/comics/conditional_risk.png

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Example 1: Conditional Probability

Roll a fair 4 sided die 3 timesA = the event that two 1’s are tossedB = the event that the first roll is an 1C = the event that the second roll is an 1

Find: P(B|A), P(A|B), P(B|C)

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Example 2: Conditional ProbabilityA bus arrives punctually at a bus stop every half hour.

Each morning, a commuter named Sarah leaves her house and casually strolls to the bus stop.

a) Find the probability that that wait time is at least 10 minutes.

b) Find the probability that the wait time is at least 10 minutes given that if someone is waiting there for more than 15 minutes, they will get a ride from a passing car.

c) Find the probability that the wait time is at most 10 minutes given that if someone is waiting there for more than 15 minutes, they will get a ride from a passing car.

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Example 3: Conditional Probability

1) From the set of all families with two children, a family is selected at random and is found to have a girl. What is the probability that the other child of the family is a girl?

2) From the set of all families with two children, a child is selected at random and is found to be a girl. What is the probability that the other child of the family is a girl?

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Theorem 4.10: Distributive Laws

For any events, A1, A2, …

j jj j

A B A B

j jj j

A B A B

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Example: Conditional Probabilities - Axioms

Assume that the probability that there will be more than 1 inch of snow next week given that the temperature rises above 30 oF is 0.3.

a) What is the probability that there will be less than 1 inch of snow next week given that the temperature rises above 30 oF?

b) Given that the probability that there will be more than 2 inches of rain given that the temperature rises above 30 oF is 0.8 and the probability that there will be more than 2 inches of rain and there will be less than 1 inch of snow given that the temperature rises above 30 oF is 0.6, what is the probability that there will be more than 2 inches of rain or there will be less than 1 inch of snow given that the temperature rises above 30 oF?

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Chapter 5: Bayes’ Theorem(And Additional Applications)

http://pactiss.org/2011/11/02/bayesian-inference-homo-bayesianis/

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Example: Bayes’ TheoremIn a bolt factory, 30, 50, and 20% of the production

is manufactured by machines I, II, and III, respectively. If 4, 5, and 3% of the output of these respective machines is defective, what is the probability that a randomly selected bolt that is found to be defective is manufactured by machine III?

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Example: Bayes’ Theorem (Monty Hall Problem)

This follows the game show ‘Let’s Make a Deal’ which was hosted by Monty Hall for many years. In the game show, there are three doors, behind each of which is one prize. Two of the prizes are worthless and the other one is valuable. A contestant selects one of the doors, following which the game show host (who does know where the valuable prize is), opens one of the remaining two doors to reveal a worthless prize. The host then offers the contestant the opportunity to change his selection. Should the contestant switch doors?

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Example: Bayes’ Theorem (Diagnostic Tests)

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?

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Examples: General Multiplication Law1) A consulting firm is awarded 51% of the contracts it bids on.

Suppose that Melissa works for a division of the firm that gets to do 25% of the projects contracted for. If Melissa directs 41% of the projects submitted to her division, what percentage of all bids submitted by the firm will result in contracts for projects directed by Melissa?

2) Supposed 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. What is the probability that we find both of the defective fuses in exactly three tests?

3) Using Pólya’s Urn, with r red balls, b black balls and c, what is the probability that the first two balls are red and the last ball is black?

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Example: Electrical ComponentsFor the following problems, assume that each switch is independently closed or open with probability p and 1 - p, respectively. Note: The answers should include ‘p’.1) In the figure below, there are 4 switches labeled 1, 2, 3 and

4. If a signal is fed to the input, what is the probability that it is transmitted to the output?

2) If a circuit is composed only of n parallel components, then what is the probability that, at a specified time, the system is working?