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**Copyright © Cengage Learning. All rights reserved.**

2 Probability Copyright © Cengage Learning. All rights reserved.

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**Probability: Definitions**

Random Experiment (RE) Any action or process whose outcome is subject to uncertainty Sample Space (S) The set of all possible outcomes of the RE

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**Sample Space What is the sample space of the following:**

if I roll one 4-sided die? if I roll two 4-sided dice? if I toss a coin until the first head appears? length of a bolt

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**Probability: Definitions (cont.)**

Outcome One of the possible results from a RE Event Any collection (subset) of outcomes contained in the sample space Types of Events Simple: the event has one outcome Compound: the event has more than one outcome

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**Events – Simple or Compound**

Are the following events considering a 4-sided die simple or compound? Let E1 be the event of rolling a 3 on the die. Let E2 be the event of rolling a number greater than 2 on a die. Let E3 be the event of rolling a 5 on the die. Let E4 be the event of rolling an odd number.

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**Events – Simple or Compound (cont)**

Are the following events considering tossing a coin until the first head appears simple or compound? Let A be the event that it takes exactly 2 tosses. Let B be the event that it takes less than two tosses. Let C be the event of making less than 2 or greater than 5 tosses. Let D be the event that it takes more than 3 tosses.

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**Set Theory Complement, A’ Union (A U B) Intersection (A ∩ B)**

Null Event, Mutually Exclusive (disjoint)

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**Set Theory: Examples Rolling a 4-sided die. Let A = {1,2}, B = {2,3}**

A U B b) A ∩ B c) A’ d) Are A and B mutually exclusive? Drawing a card from a deck of cards, A = red card, B = black card a) Are A and B mutually exclusive? Tossing a coin until the first head A = {TH} B = {H,TH,TTH] C = {TH,TTH,TTTTH} D = {TTTH,TTTTH,TTTTTH,…} a) A U B U C b) B ∩ C c) C ∩ D d) B’ e) A ∩ (B U C) f) (B ∩ C) U A

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**Probability: Axioms For any event A, P(A) ≥ 0 P(S) = 1**

If A1, A2, … is an countably infinite collection of disjoint events, then 𝑃 𝐴 1 ∪ 𝐴 2 ∪⋯ = 𝑖=1 ∞ 𝑃( 𝐴 𝑖 )

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**Probability: Propositions**

If A1, A2, …, An is a finite collection of disjoint events, then 𝑃 𝐴 1 ∪ 𝐴 2 ∪⋯∪ 𝐴 𝑛 = 𝑖=1 𝑛 𝑃( 𝐴 𝑖 ) For any event A, P(A) = 1 – P(A’) Example: Roll two 4-sided dice. What is the probability that the sum of the two dice is greater than 2?

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**Probability: Propositions 2**

For any event A, P(A) ≤ 1 For any two events A1 and A2, P(A1 U A2) = P(A1) + P(A2) – P(A1 ∩ A2) Example: A card is drawn from a well-shuffled deck of cards. What is the probability that the card is an ace or a diamond?

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**Interpreting Probability**

Let A be the event that a package sent within the state of California for 2nd day delivery actually arrives within one day.

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**Equally Likely Outcomes**

Rolling two 4-sided dice. What is the probability that a 1 will be rolled on the red die? Draw one card from a deck of cards. What is the probability of drawing a King? What is the probability of drawing a heart? A song is chosen at random from a person’s mp3 player. If there are 10 alternative, 5 blues, 7 jazz and 25 rock songs on the player, what is the probability that a rock song is played?

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**Counting Techniques ordered unordered With replacement**

ordered unordered With replacement Without replacement

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Product Rule Requirements: ordered, with replacement Example: How many pairs are there when you roll a 4-sided die and a 3-sided die? Calculation: If the first element of an ordered k-tuple can be selected in n1 ways, the second element in n2 ways up to the kth element in nk ways, then there are 𝑛 1 𝑛 2 ⋯ 𝑛 𝑘 possible k-tuples.

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**Product Rule: Examples**

How many pairs are there when you roll a 4-sided die and a 3-sided die? How many pairs are possible for the following situation: Roll a 4-sided die with a result of n. Then roll a 2n-sided die. [This can not be done via the product rule.] How many different possibilities are there when you rolls a 4-sided die, a 3-sided die and a 10-sided die?

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**Counting Techniques ordered unordered With replacement Product Rule**

ordered unordered With replacement Product Rule (multiplication, tree diagrams) Without replacement

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Permutation Requirements: ordered, without replacement Example: How many different ways can we draw 4 cards from the 13 spades in the deck of cards without replacement? Calculation: 𝑃 𝑘,𝑛 = 𝑛! 𝑛−𝑘 !

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**Permutation: Examples**

How many different ways can we draw 4 cards from the 13 spades in the deck of cards without replacement? In a horse race consisting of 10 horses, how many different ways are there to choose the horses that finish first, second and third? If there are 20 students in a club, how many unique ways can the president, vice president, secretary and treasurer be elected?

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**Counting Techniques ordered unordered With replacement Product Rule**

ordered unordered With replacement Product Rule (multiplication, tree diagrams) Without replacement Permutation 𝑃 𝑘,𝑛 = 𝑛! 𝑛−𝑘 !

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Combination Requirements: unordered, without replacement Example: How many ways can I choose 3 of the 6 different type of dark red dice? Calculation: 𝑛 𝑘 =𝐶 𝑘,𝑛 = 𝑛! 𝑘! 𝑛−𝑘 !

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Combination How many ways can I choose 3 of the 6 different type of dark red dice? In a horse race consisting of 10 horses, how many different ways are there to choose the horses that finish in the money? If there are 20 students in a club, how many unique ways can you choose for four students to go to a conference?

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**Counting Techniques ordered unordered With replacement Product Rule**

ordered unordered With replacement Product Rule (multiplication, tree diagrams) Without replacement Permutation 𝑃 𝑘,𝑛 = 𝑛! 𝑛−𝑘 ! Combination 𝑛 𝑘 =𝐶 𝑘,𝑛 = 𝑛! 𝑘! 𝑛−𝑘 !

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**Probability - Counting**

I have 10 dice in the back, 6 dark red, 1 red and 3 white. What is the probability that the 3 dice are: 1 white and 2 dark red? all red?

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Counting - Procedure Construct the sample space where each outcome is equally likely to happen. Determine the event of interest. Count the outcomes of the event and the total number of outcomes. Find the probability by: 𝑃 𝐴 = 𝑁(𝐴) 𝑁

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**Examples for Counting Techniques**

Roll two 8-sided dice. What is the probability that the sum of the two numbers is 5? (0.0625) Draw two cards from a suit of 13 cards (say diamonds), what is the probability that the sum of the two cards is even? (A = 1, J = 11, Q = 12, K = 13)? (0.462) The IRS decides that it will audit the returns of 3 people from a group of 18. If 8 of the people are women, what is the probability that all 3 of people audited are women? (0.0686) Arizona places consist of three digits followed by three letters. What is the probability that a particular license plate doesn’t have any repeating digits or letters? (0.639)

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**Conditional Probability**

If we draw a card from a deck of 52 cards, what is the probability of getting a heart? Assuming that the card is not a club, what is the probability of getting a heart?

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**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. What is the probability that we find both of the defective fuses in the first two tests? What is the probability that the first tested fuse is good out of 3 fuses and the last two tested are defective?

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**Example Bayes’ Law: Coins/Balls**

Flip a coin. If the coin is a head, draw a ball from a bag with 2 white and 2 black balls. If the coin is a tail, draw a ball from a bag with 1 white and 2 black balls. If the drawn ball is white, what is the probability that the coin was a head?

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**Example Bayes’ Law: Test for Rare Disease Test**

Suppose we know that 0.1% of the total population has Dercum’s disease. If a person has the disease, the test will successfully detect it with 95% accuracy, and doesn’t detect it 5% of the time. If a person does not have the disease, the test will be incorrect 1% of the time. What is the probability that when the test shows a person has Dercum’s disease, the person really has the disease?

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**Example Bayes’ Law: Light Bulbs**

A store stocks light bulbs from three suppliers. Suppliers A, B, and C supply 10%, 20%, and 70% of the bulbs, respectively. It has been determined that company A’s bulbs are 1% defective while company B’s are 3% defective and company C’s are 4 % defective. If a bulb is selected at random and found to be defective, what is the probability that it came from supplier B?

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**Example: Independence (1)**

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.

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**Example: Independence (2)**

All components in a the network function independently. The probabilities of success are as follows: P(A) = 0.9, P(B) = 0.92, P(C) = 0.8, P(D) = What is the probability that the network will fail?

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**Mutually Exclusive vs. Independent**

In each situation, are the following two events a) mutually exclusive 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

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