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**Probability and Simulation: The Study of Randomness**

Chapter 6 Probability and Simulation: The Study of Randomness

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**6.1 Objectives Students will be able to: Define Simulation.**

List the five steps involved in a simulation. Explain what is meant by independent trials. Use a table of random digits to carry out a simulation. Given a probability problem, conduct a simulation in order to estimate the probability desired. Use a calculator or a computer to conduct a simulation of a probability problem.

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6.1 Simulation The imitation of chance behavior, based on a model that accurately reflects the phenomenon under consideration, is called simulation.

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Simulation Steps Step 1: State the problem or describe the random phenomenon Toss a coin 10 times. What is the likelihood of a run of at least 3 consecutive heads or 3 consecutive tails? Step 2: State the assumptions. There are two: A head or tail is equally likely to occur on each toss. Tosses are independent of each other (that is, what happens on one toss will not influence the next toss)

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**Simulation Steps Step 3: Assign digits to represent outcomes.**

One digit simulates one toss of the coin. Odd digits represent heads; even digits represents tails. Step 4: Simulate many repetitions. We will complete 25 repetitions for this simulation. In a random number table, even and odd digits occur with the same long-term relative frequency, 50%. This is just one assignment of digits for coin tossing. Successive digits in the table simulate independent tosses.

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**Simulation Steps Step 5: State your conclusions.**

We estimate the probability of a run of size 3 by the proportion Estimated probability = 23/25 = 0.92. 25 repetitions is not enough to be confident that our estimate is accurate. We can use the computer to do thousands of trials for us. A long simulation (mathematical analysis) finds that the true probability is about

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Assigning Digits Choose a person at random from a group of which 70% are employed. 0, 1, 2, 3, 4, 5, 6 = employed 7, 8, 9 not employed 00, 01, …, 69 employed 70, 71,…, 99 not employed Choose a person at random from a group of which 73% are employed. 00, 01, …, 72 employed 73, 74, …, 99 not employed 0, 1, 2, 3, 4, 5, 6 = employed 7, 8, 9 not employed 00, 01, …, 69 employed 70, 71,…, 99 not employed

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Assigning Digits Choose a person at random from a group of which 50% are employed, 20% are unemployed, and 30% are not in the labor force. 0, 1, 2, 3, 4 = employed 5, 6 unemployed 7, 8, 9 not in the labor force 0,1 = unemployed 2, 3, 4 = not in the labor force 5, 6, 7, 8, 9, = employed 0, 1, 2, 3, 4 = employed 5, 6 unemployed 7, 8, 9 not in the labor force 0,1 = unemployed 2, 3, 4 = not in the labor force 5, 6, 7, 8, 9, = employed

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**Example Page 397 #6.1 Establishing Correspondence**

State how you would use the following aids to establish a correspondence in a simulation that involves a 75% chance: A coin A six-sided dice A random digit table A standard deck of playing cards a. Flip the coin twice. Let HH represent failure, and let the other outcomes (HT, TH, TT) represent success. Let 1, 2, 3, represent a success, and let 4 represent a failure. If 5 or 6 come up, ignore them and roll again. Peel off two consecutive digits from the table; let 00 through 74 represent a success, and let 75 through 99 represent a failure. Let diamonds, spades, and clubs represent a success, and let hearts represent a failure.

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**Example Page 404 #6.15 The birthday problem**

Use your calculator and a simulation method to determine the chances of at least 2 students with the same birthday in a class of 23 unrelated students. Determine the chances of at least 2 people having the same birthday in a room of 41 people. What assumptions are you making in your simulations?

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Types of Simulations Situations in which our interest is in the number of successes out of a fixed number of trials (assuming equal probabilities and independence from trial to trial) are often solved using the binomial distribution. Situations in which our interest is in how many trials it takes for an event to occur (again assuming equal probabilities and independence from trial to trial) are often solved using the geometric distribution.

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**6.2 Objectives Students will be able to**

Explain how the behavior of a chance event differs in the short and long run. Explain what is meant by a random phenomenon. Explain what it means to say that the idea of probability is empirical. Define probability in terms of relative frequency. Define sample space. Define event. Explain what is meant by probability model. Construct a tree diagram. Use the multiplication principle to determine the number of outcomes in a sample space. Explain what is meant by sampling with replacement and sampling without replacement.

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6.2 Objectives List the four rules that must be true for any assignment of probabilities. Explain what is meant by {A U B} and {A ∩ B}. Explain what is meant by each of the regions in a Venn diagram Give an example of two events A and B where A ∩ B = Ø. Use a Venn diagram to illustrate the intersection of two events A and B. Compute the probability of an event, given the probabilities of the outcomes that make up the event. Explain what is meant by equally likely outcomes. Compute the probability of an event in the cases of equally likely outcomes. Define what it means for two events to be independent. Give the multiplication rule for independent events. Given two events, determine if they are independent.

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**Randomness 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 random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. That is, the probability is long-term relative frequency.

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**Types of Probability Theoretical probability (Classical)**

Empirical probability is based on observations rather than theorizing. Subjective probability

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**Example Page 410 #6.21 Pennies Spinning**

Hold a penny upright on its edge under your forefinger on a hard surface, then snap it with your forefinger so that it spins for some time before falling. Based on 50 spins, estimate the probability of heads.

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Probability Models The sample space is the set of all possible outcomes. An event is any 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 mathematical description of a random phenomenon consisting of two parts: A sample space, S and A way of assigning probabilities to events.

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**Types of models Discrete models have a countable number of outcomes.**

Continuous models correspond to intervals on the number line. Discrete: number of heads observed on a flip of 3 coins Continuous: heights of a sample of 15 ninth-graders

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**There are 36 possible outcomes.**

Tree Diagrams Two dice are rolled. Describe the sample space. Start 1st roll 2nd roll 1 2 3 4 5 6 There are 36 possible outcomes.

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**Two dice are rolled and the sum is noted.**

1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6

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**Find the probability the sum is 4.**

Find the probability the sum is 4 or 11.

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**Multiplication Principle**

If you can do one task in n1 ways and a second task in n2 ways, then both tasks can be done in n1 x n2 number of ways. How many ways can you flip 3 coins? How many ways can you flip a coin and roll a die?

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**Replacement Sampling with replacement Sampling without replacement**

How many 4 digit pin numbers can you make? How many 4 digit pin numbers can you make if all numbers are distinct?

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**Example Number of ways Sum Outcomes 1 2 1, 1 3 1,2 2,1 4 5 6 7 8 9 10**

11 12 Page 417 #6.35 Rolling Two Dice In how many ways can you get an even sum? In how many ways can you get a sum of 5? Of 8? Describe a pattern you see in the table.

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**Probability Rules Rule 1: Any probability is a number between 0 and 1.**

Rule 2: The sum of the probabilities of all possible outcomes must equal 1. Rule 3: If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. Rule 4: The probability that an event will occur is 1 minus the probability that the even does occur.

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**Venn diagrams Mutually exclusive (disjoint) Union (A U B)**

Intersect (A ∩ B)

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Venn diagrams Empty Set Complement A U Ac = S A and Ac = empty set

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Equally Likely If a random phenomenon has k possible outcomes that are all equally likely, then each individual outcome has probability 1/k.

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**Example Page 423 #6.38 Distribution of M&M colors**

If you draw an M&M candy at random from a bag of candies, the candy you draw will have one of six colors. The probability of drawing each color depends on the proportion of each color amoung the candies made. The table below gives the probability of each color for a randomly chosen milk chocolate M&M: What must be the probability of drawing a blue candy? Color: Brown Red Yellow Green Orange Blue Probability: 0.13 0.14 0.16 0.20 ?

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**The probabilities for peanut M&M’s are different.**

What is the probability that a peanut M&M is blue? What is the probability that a milk chocolate M&M is red, yellow or orange? What is the probability that a peanut M&M is one of these colors? Color: Brown Red Yellow Green Orange Blue Probability: 0.12 0.15 0.23 ?

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**Multiplication Rule for Independent Events**

Rule 5: Two events 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) **A and B are independent iff P(A|B) = P(A) Independence and disjoint are not the same thing… if two events are disjoint then they cannot be independent because knowing that one occurred would tell us that the other cannot.

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Caution… The multiplication rule applies only to independent events. You cannot use it if events are not independent! The addition rule applies only to disjoint events. Disjoint does not mean independent.

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**Example Page 430 #6.46 Defective Chips**

An automobile manufacturer buys computer chips from a supplier. The supplier sends a shipment containing 5% defective chips. Each chip chosen from the shipment has probability 0.05 of being defective, and each automobile uses 12 chips selected independently. What is the probability that all 12 chips in a car will work properly?

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**Example Page 430 #6.48 College Student Demographics**

Choose a random college student at least 15 years of age. We are interested in the events A = {The person chosen is male} B = {The person chosen is 25 years or older} Government data recorded in Table 4.5 (page 292) allow us to assign probabilities to these events. Explain why P(A) = 0.44 Find P(B) Find the probability that the person chosen is a male at least 25 years old, P(A and B). Are these events A and B independent?

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**Example Page 430 #6.51 Telephone Success**

Most sample surveys use random digit dialing equipment to call residential telephone numbers at random. The telephone polling firm Zogby International reports that the probability that a call reaches a live person is Calls are independent. A polling firm places 5 calls. What is the probability that none of them reaches a person? When calls are made to New York City, the probability of reaching a person is What is the probability that none of 5 calls made to New York City reaches a person?

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**Example Page 432 #6.53 Student Survey**

Choose a student at random from a large statistics class. Give a reasonable sample space S for answers to each of the following questions. (In some cases you may have the freedom to specify S.) Are you right or left handed? What is your height in centimeters? (1 inch = 2.54 cm) How much money in coins (not bills) are you carrying? How many minutes did you study last night?

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**Example Page 434 #6.62 Roulette**

A roulette wheel has 38 slots, numbers 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet various combinations of numbers and colors. What is the probability that the ball will land in any one slot? If you bet on “red”, you win if the ball lands in a red slot. What is the probability of winning? The slot numbers are laid out on a board on which gamblers place their bets. One column of numbers on the board contains multiples of 3, that is 3, 6, 9, …, 36. You place a ”column bet” that wins if any of these numbers comes up. What is your probability of winning?

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**6.3 Objectives Students will be able to**

State the Addition Rule for disjoint events. State the general addition rule for union of two sets. Given two events A and B, compute P(A U B). Define what is meant by a joint event and joint probability. Given two events, compute their joint probability. Explain what is meant by the conditional probability P(A|B). State the general multiplication rule to define P(B|A). Explain what is meant by Bayes’s rule. Define independent events in terms of a conditional probability.

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Union The Union of any collection of events is the event that at least one of the collection occurs.

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**Addition Rule for Disjoint Sets**

If events A, B, and C are disjoint, then P(A or B or C) = P(A) + P(B) + P(C)

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**General Addition Rule for Unions of Two Events**

P(A or B) = P(A) + P(B) – P(A and B)

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**Example Page 441 #6.70 Tastes in Music I**

Musical styles other than rock and pop are becoming more popular. A survey of college students finds that 40% like country music, 30% like gospel music, and 10% like both. Make a Venn diagram with these results. What percent of college students like country but not gospel? What percent like neither country nor gospel?

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**Conditional Probability**

P(A|B) “probability of A, given B”

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**General Multiplication Rule**

The joint probability that events A and B will both happen can be found by P(A and B) = P(A) x P(B|A) Here P(B|A) is the conditional probability that B occurs, given the information that A occurs.

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**Conditional Probability**

When P(A) > 0, the conditional probability of B, given A, is `P(B|A) = P(A and B) P(A) We require that the probability of A occurring be greater than 0 because the probability of B given A makes no sense if A cannon occur. Notice this comes from rearranging the multiplication rule

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**Example Page 446 #6.72 Pay at the Pump**

At a self-service gas station, 40% of the customers pump regular gas, 35% pump midgrade, and 25% pump premium gas. Of those who pump regular, 30% pay at least $20. Of those who pump midgrade, 50% pay at least $20. And of those who pump premium, 60% pay at least $20. What is the probability that the next customer pays at least $20?

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**Example Page 447 #6.76 The probability of a flush**

A poker player holds a flush when all 5 cards in the hand belong to the same suit. We will find the probability of a flush when all the cards are dealt. Remember that a deck contains 52 cards,13 of each suit, and that when the deck is well shuffled, each card dealt is equally likely to be any of those that remain in the deck. We will concentrate on spades, what is the probability that the first card dealt is a spade? What is the conditional probability that the second card is a spade, given that the first is a spade?

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Continue to count the remaining cards to find the conditional probabilities of a spade on the third, fourth, and fifth card, given in each case that all previous cards are spades. The probability of being dealt 5 spades is the product of the five probabilities that you have found. Why? What is this probability? The probability of being dealt 5 hearts or 5 diamonds or 5 clubs is the same as the probability of being dealt 5 spades. What is the probability of being dealt a flush?

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Intersection The intersection of any collection of events is the probability that all of the events occur.

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Example Page 448 Example 6.30

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Independent Events Two events A and B that both have positive probability are independent if P(B|A) = P(B)

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