Chapter 6 Probability and Simulation 6.1 Simulation.
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Transcript of Chapter 6 Probability and Simulation 6.1 Simulation.
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Chapter 6 Probability and Simulation
6.1 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
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
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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 = 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
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Various Simulation Scenarios
• Example 6.4 – page 395 - 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 .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 .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
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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: 6.17 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 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)
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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