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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. State the problem or describe the experiment

Chapter 6 Probability and Simulation

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Chapter 6 Probability and Simulation

6.1 Simulation

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

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

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

- 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).

- 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.

- 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

- 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

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

- 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.

- 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

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

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

Chapter 6 Probability and Simulation

6.2 Probability Models

- 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)

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

- 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

- 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 repetition.

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.

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}

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

- 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.

- 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.

- 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}

- Example: a coin is tossed 4 times. Then “exactly 2 heads” is an event.

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

Example 6.13 on page 419

- 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

- 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

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

- 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

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)

- 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)

- Example 6.25, page 442, 443

- 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

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

- 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 6.30, Page 448-9, Online Chatrooms

- Example 6.31, page 450, Chat Room Participants

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

P(B | A ) = P(B)

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