Chapter 6 Probability and Simulation

6.1 Simulation

Simulation

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

Steps for Conducting a Simulation

1. State the problem or describe the experiment

2. State the assumptions

3. Assign digits to represent outcomes

4. Simulate many repetitions

5. State your conclusions

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?

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).

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.

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

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 = 92.25

23

If we wrote a computer simulation program and ran many thousands of repetitions you would find that the true probability is about .826

Various Simulation Scenarios

• Example 6.4 – page 395 - Choose one person at random from a group of 70% employed. Simulate using random number table.

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.

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

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)

Homework

• Read 6.1, 6.2

• Complete Problems 1-4, 8, 9, 12

Chapter 6 Probability and Simulation

6.2 Probability Models

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)

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

In the long run, the proportion of heads approaches .5, the

probability of a head

Researchers with Time on their Hands

• French Naturalist Count Buffon (1707 – 1788) tossed a coin 4040 time. Results: 2048 head or a proportion of .5069.

• English Statistictian Karl Person 24,000 times. Results 12, 012, a proportion of .5005.

• Austrailian mathematician and WWII POW John Kerrich tossed a coin 10,000 times. Results 5067 heads, proportion of heads .5067

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

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.

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.

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}

The 36 Possible Outcomes in rolling two dice.

A Tree Diagram can help you understand all the possible outcomes

in a Sample Space of Flipping a coing and rolling one die.

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

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.

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.

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}

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

Venn diagram showing disjoint Events A and B

Venn diagram showing the complement Ac of an event A

Complement Example

Example 6.13 on page 419

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

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

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: 6.17 page 426

Homework

• Read Section 6.3

• Exercises 22, 24, 28, 29, 32-33, 36, 38, 44

Probability And Simulation: The Study of Randomness

6.3 General Probability Rules

Rules of Probability Recap

Rule 1. 0 < P(A) < 1 for any event ARule 2. P(S) = 1Rule 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)

Key Term

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

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)

General Rule for Unions of Two Events,

P(A or B) = P(A) + P(B) – P(A and B)

Example 6.23, page 438

Conditional Probability

• Example 6.25, page 442, 443

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

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

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)

Example 6.29, page 448: Extended Multiplication Rule

Tree Diagrams Revisted

• Example 6.30, Page 448-9, Online Chatrooms

Bayes’s Rule

• Example 6.31, page 450, Chat Room Participants

Independence Again

Two events A and B that both have positive probability are independent if

P(B | A ) = P(B)

Homework

• Exercises #71-78, 82, 86-88