Transcript 6-1 simulation

Unit 6
Probability & Simulation:
the Study of randomness
Simulation
Probability Models
 General Probability Rules
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Questions of chance
Successive coin tosses
Having a girl among my four children
Cereal Box Sports cards
Airline Overbookings
Answering questions of chance
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Estimate the likelihood by actually observing the
phenomenon many times.
Develop a probability model and calculate a
theoretical answer.
Simulate multiple repetitions.
Simulation
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On the AP Exam, you may receive full
credit for solving a probability problem
by simulation even if the problem does
not call for it!
2 Types of Simulation
1. Simulation with a Fixed Number of
Trials
Number of trials determined by the
problem.
Binomial probability
2. Wait-time Simulation
We continue to simulate until a certain
condition is met.
Fixed Number of Trials
Simulation
Ex 6.3 Coin Tossed Ten Times
1. State the problem or describe the
random phenomenon.
2. State the assumptions. (individual
likeliness, independent trials)
3. Assign digits to represent outcomes.
Are repeated digits okay?
4. Simulate many repetitions.
5. State your conclusions.
Wait-time Simulation
Ex. 6.6 Girl or Four
1. State the problem or describe the random
phenomenon. State stopping rule.
2. State the assumptions. (individual
likeliness, independent trials)
3. Assign digits to represent outcomes. Are
repeated digits okay?
4. Simulate many repetitions.
5. State your conclusions.
Section 6.2 Probability Models
Chance behavior is unpredictable in the
short run but has a regular and
predictable pattern in the long run.
Probability is used to predict future
behavior.
The idea of probability
Coin tossing
Count Buffon
in the 1700's
4040 times
2048 heads for
0.5069
John Kerrich
imprisoned in WWII
10,000 times
5067 heads for 0.5067
Randomness
Probability theory is the branch of
mathematics that describes random
behavior.
1.Random does not mean haphazard.
2. You must have a long series of
independent trials.
3. Probability is empirical, meaning it is
based on observation of many trials
rather than theorizing.
4.Computer simulations are useful as
they provide many trials, very fast.
Sample Space
Being able to properly enumerate the outcomes in
a sample space is critical to determining
probabilities.
Three techniques to make sure you don’t
accidentally overlook any outcomes:
1.Tree diagram
2.Multiplication principle
3.Organized or (Systematic) List
Tree Diagram
Sometimes it's helpful to use a tree diagram.
Tree diagrams are good for relatively small
sample spaces.
What is the sample space for tossing a coin and
rolling a die?
Tree Diagram
Multiplication Principle or
Fundamental Counting Principle
What is the sample space when we record four
coin tosses in order?
Organized List
What is the sample space when we roll
two dice?
Sample Space
Sometimes the sample space varies depending
on what exactly you're asking for!
What is the sample space when we want to
know the number of heads in four coin tosses?
Discrete vs. Continuous Models
Discrete models have a countable number of
outcomes. Ex: Flipping a coin
Continuous models have an infinite number of
outcomes. They correspond to intervals on the
number line.
Ex: heights of a sample of 15 ninth-graders
Ex: all the numbers between 0 and 1
S = {all numbers between 0 and 1}
Sampling With and Without
Replacement
Sampling With Replacement: the same outcome
can occur more than once in a sample
Ex: Draw 5 numbers from a hat, after each number
is drawn, put the number back in the hat.
Sampling Without Replacement: the same
outcome cannot occur more than once in a sample
Ex: Draw 5 numbers from a hat, after each number
is drawn, do not put the number back in the hat.
Sampling With and Without
Replacement
How many 3 digit numbers can you make with
replacement?
How many 3 digit numbers can you make without
replacement?
Use the context of the problem to determine whether
the selection is with or without replacement.
Counting Problems
How many ways can you roll two dice and get a
sum of 3? What are the outcomes?
How many ways can you roll two dice and get a
sum of 7? What are the outcomes?