Transcript Chapter 6 Probability and Simulation

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 = 23
25
 .92
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.
Rule 2.
Rule 3.
Rule 4.
Rule 5.
0 < P(A) < 1 for any event A
P(S) = 1
Addition rule: If A and B are disjoint
events, then
P(A or B) = P(A) + P(B)
Complement rule: For any event A,
P(Ac) = 1 – P(A)
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