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**Chapter 6 Probability and Simulation**

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

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**Steps for Conducting a Simulation**

State the problem or describe the experiment State the assumptions Assign digits to represent outcomes Simulate many repetitions State your conclusions

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**Step 1: State the problem or describe the experiment**

Toss a coin 10 times. What is the likelihood of a run of at least 3 consecutive heads or 3 consecutive tails?

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**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 (ie: what happens on one toss will not influence the next toss).

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**Step 3 Assign Digits to represent outcomes**

Since each outcome is just as likely as the other, and there you are just as likely to get an even number as an odd number in a random number table or using a random number generator, assign heads odds and tails evens.

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**Step 4 Simulate many repetitions**

Looking at 10 consecutive digits in Table B (or generating 10 random numbers) simulates one repetition. Read many groups of 10 digits from the table to simulate many repetitions. Keep track of whether or not the event we want ( a run of 3 heads or 3 tails) occurs on each repetition. Example 6.3 on page 394

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Step 5 State your conclusions. We estimate the probability of a run by the proportion Starting with line 101 of Table B and doing 25 repetitions; 23 of them did have a run of 3 or more heads or tails. Therefore estimate probability = If we wrote a computer simulation program and ran many thousands of repetitions you would find that the true probability is about .826

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**Various Simulation Scenarios**

Example 6.4 – page Choose one person at random from a group of 70% employed. Simulate using random number table.

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Frozen Yogurt Sales Example 6.5 – page 396 – Using random number table simulate the flavor choice of 10 customers entering shop given historic sales of 38% chocolate, 42% vanilla, 20% strawberry.

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A Girl or Four Example 6.6 – Page 396 – Use Random number table to simulate a couple have children until 1 is a girl or have four children. Perform 14 Simulation

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**Simulation with Calculator**

Activity 6B – page 399 – Simulate the random firing of 10 Salespeople where 24% of the sales force are age 55 or above. (20 repetitions)

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Homework Read 6.1, 6.2 Complete Problems 1-4, 8, 9, 12

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**Chapter 6 Probability and Simulation**

6.2 Probability Models

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Key Term Probability is the branch of mathematics that describes the pattern of chance outcomes (ie: roll of dice, flip of coin, gender of baby, spin of roulette wheel)

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Key Concept “Random” in statistics is not a synonym of “haphazard” but a description of a kind of order that emerges only in the long run

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**In the long run, the proportion of heads approaches**

In the long run, the proportion of heads approaches .5, the probability of a head

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**Researchers with Time on their Hands**

French Naturalist Count Buffon (1707 – 1788) tossed a coin 4040 time. Results: 2048 head or a proportion of English Statistictian Karl Person 24,000 times. Results 12, 012, a proportion of Austrailian mathematician and WWII POW John Kerrich tossed a coin 10,000 times. Results 5067 heads, proportion of heads .5067

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Key Term / Concept 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

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Key Term / Concept The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetition.

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**Key Term / Concept As you explore randomness, remember**

You must have a long series of independent trials. (The outcome of one trial must not influence the outcome of any other trial) We can estimate a real-world probability only by observing many trials. Computer Simulations are very useful because we need long runs of data.

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Key Term / Concept The sample space S of a random phenomenon is the set of all possible outcomes. Example: The sample space for a toss of a coin. S = {heads, tails}

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**The 36 Possible Outcomes in rolling two dice.**

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A Tree Diagram can help you understand all the possible outcomes in a Sample Space of Flipping a coing and rolling one die.

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Key Concept Multiplication Principle - If you can do one task in n1 number of ways and a second task in n2 number of ways, then both tasks can be done in n1 x n2 number of ways. ie: flipping a coin and rolling a die, 2 x 6 = 12 different possible outcomes

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Key Term / Concept With Replacement – Draw a ball out of bag. Observe the ball. Then return ball to bag. Without Replacement – Draw a ball out of bag. Observe the ball. The ball is not returned to bag.

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**Key Term / Concept With Replacement – Three Digit number**

10 x 10 x 10 = 1000 ie: lottery select 1 ball from each of 3 different containers of 10 balls Without Replacement – Three Digit number 10 x 9 x 8 = 720 ie: lottery select 3 balls from one container of 10 balls.

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Key Concept / Term An event is an outcome or a set of outcomes of a random phenomenon. An event is a subset of the sample space. Example: a coin is tossed 4 times. Then “exactly 2 heads” is an event. S = {HHHH, HHHT,………..,TTTH, TTTT} A = {HHTT, HTHT, HTTH, THHT, THTH, TTHH}

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**Key Definitions Sometimes we use set notation to describe events.**

Union: A U B meaning A or B Intersect: A ∩ B meaning A and B Empty Event: Ø meaning the event has no outcomes in it. If two events are disjoint (mutually exclusive), we can write A ∩ B = Ø

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**Venn diagram showing disjoint Events A and B**

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**Venn diagram showing the complement Ac of an event A**

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Complement Example Example 6.13 on page 419

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**Probabilities in a Finite Sample Space**

Assign a Probability to each individual outcome. The probabilities must be numbers between 0 and 1 and must have a sum 1. The probability of any event is the sum of the outcomes making up the event Example 6.14 page 420

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**Assigning Probabilities: equally likely outcomes**

If a random phenomenon has k possible outcomes, all equally likely, then each individual outcome has probability 1/k. The probability of any event A is P(A) = count of outcomes in A count of outcomes in S Example: Dice, random digits…etc

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**The Multiplication Rule for Independent Events**

Rule 3. 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) Examples: page 426

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Homework Read Section 6.3 Exercises 22, 24, 28, 29, 32-33, 36, 38, 44

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

6.3 General Probability Rules

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**Rules of Probability Recap**

Rule 1. 0 < P(A) < 1 for any event A Rule 2. P(S) = 1 Rule 3. Addition rule: If A and B are disjoint events, then P(A or B) = P(A) + P(B) Rule 4. Complement rule: For any event A, P(Ac) = 1 – P(A) Rule 5. Multiplication rule: If A and B are independent events, then P(A and B) = P(A)P(B)

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Key Term The union of any collection of events is the event that at least one of the collection occurs.

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The addition rule for disjoint events: P(A or B or C) = P(A) + P(B) + P(C) when A, B, and C are disjoint (no two events have outcomes in common)

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**General Rule for Unions of Two Events, P(A or B) = P(A) + P(B) – P(A and B)**

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Example 6.23, page 438

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**Conditional Probability**

Example 6.25, page 442, 443

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**General Multiplication Rule**

The joint probability that both of two events A and B happen together can be found by P(A and B) = P(A)P(B | A) P(A ∩ B) = P(A)P(B | A) Example: 6.26, page 444

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**Definition of Conditional Probability**

When P(A) > 0, the conditional probability of B given A is P(B | A) = P(A and B) P(A) Example 6.28, page 445

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**Key Concept: Extended Multiplication Rule**

The intersection of any collection of events is the even that all of the events occur. Example: P(A and B and C) = P(A)P(B | A)P(C | A and B)

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**Example 6.29, page 448: Extended Multiplication Rule**

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**Tree Diagrams Revisted**

Example 6.30, Page 448-9, Online Chatrooms

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Bayes’s Rule Example 6.31, page 450, Chat Room Participants

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

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Homework Exercises #71-78, 82, 86-88

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Special Topics. Definitions Random (not haphazard): A phenomenon or trial is said to be random if individual outcomes are uncertain but the long-term.

Special Topics. Definitions Random (not haphazard): A phenomenon or trial is said to be random if individual outcomes are uncertain but the long-term.

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