Transcript SIA_Ch_5.1_Notes

Section 5.1 - Constructing Models of Random Behavior
Objectives:
1.
Build probability models by observing data
2.
Build probability models by constructing a sample
space of equally likely outcomes (symmetry)
3.
See how the Law of Large Numbers relates data to
probability
Section 5.1 - Constructing Models of Random Behavior
Fundamental Facts About Probability
• An event A is a set of possible outcomes from a random
situation.
• Probability is a number between 0 and 1 (or between 0%
and 100%) that tells how likely it is for an event to happen.
• Events that can’t happen have probability 0.
• Events that are certain to happen have probability 1.
• The probability that event A happens is denoted P(A);
the probability that event A doesn’t happen is denoted
P(A’) = 1 - P(A).
The event A’ is called the complement of A.
Section 5.1 - Constructing Models of Random Behavior
Fundamental Facts About Probability
Example: Rolling a six-sided die.
• Outcomes: {1, 2, 3, 4, 5, 6}
• Event: “rolling a even number” {2, 4, 6}
• Probabilities:
• P(1) = … = P(6) = 1/6
• P(even) = 1/2
• P(1 or 2 or 3 or 4 or 5 or 6) = 1 (certain event)
• P(7) = 0 (impossible event)
• P(2 or 3 or 4 or 5 or 6) = 1 - P(1) = 5/6 (complement)
Section 5.1 - Constructing Models of Random Behavior
Fundamental Facts About Probability
• If you have a list of all possible outcomes and all
outcomes are equally likely, then the probability of a
specific outcome is
1
the number of equally likely outcomes
and the probability of an event is
the number of outcomes in the event
the number of equally likely outcomes
Section 5.1 - Constructing Models of Random Behavior
Fundamental Facts About Probability
Example: Rolling a six-sided die
• All possible outcomes: {1, 2, 3, 4, 5, 6}
1
P(1)  P(2)  P(3)  P(4)  P(5)  P(6)   0.167
6
• Let A be the event “rolling an even number”
3 1
P(A)    0.5
6 2
Section 5.1 - Constructing Models of Random Behavior
Probability Distributions
A probability distribution gives all possible values resulting
from a random process and the probability of each.
Example: Flip a “fair coin” twice. What is the probability of 0, 1, or 2
heads?
• Outcomes: HH HT TH TT
• P(HH) = P(HT) = P(TH) = P(TT) = 1/4
• The probability distribution corresponding to the random
process of flipping a fair coin twice is:
Number of Heads
Probability
0
1/4
1
1/2
2
1/4
Section 5.1 - Constructing Models of Random Behavior
Where Do Probabilities Come From?
Observed data (long-run relative frequencies)
• Observation of thousands of births has shown that about
51% of newborns are boys. P(boy) ≈ 0.51
Symmetry (equally likely outcomes)
• Flipping a coin. Symmetry suggests that heads and tails
are equally likely. P(heads) = P(tails) = 0.5
Subjective estimates (may be based on data)
• What is the probability that Tom will be accepted into his
first-choice college?
Section 5.1 - Constructing Models of Random Behavior
Sample Spaces
A sample space for a chance process is a complete list of
disjoint outcomes. All of the outcomes in a sample space
must have a total probability equal to 1.
Disjoint means that two different outcomes can’t occur on the
same opportunity. The term mutually exclusive is sometimes
used instead of disjoint.
Section 5.1 - Constructing Models of Random Behavior
Sample Spaces
Example: Rolling a six-sided die.
•
•
Sample space (a complete list of disjoint outcomes)
{1, 2, 3, 4, 5, 6}
P(1)  P(2)  P(3)  P(4)  P(5)  P(6)  1
•
{odd, even}
1
P(odd)  P(even) 
2
P(odd)  1  P(even)
P(odd and even)  0
Section 5.1 - Constructing Models of Random Behavior
Data and Symmetry
How can you tell if the outcomes in your sample space are
equally likely?
Compare your model’s predictions with the actual results to
see if you have a good fit.
Example: Rolling a six-sided die
• The only thing that makes one side different from another is the number
of dots
• It seems unlikely that the number of dots would have much of an effect
on the probability.
• Verify by rolling the die many times. See if the actual results match the
model.
Section 5.1 - Constructing Models of Random Behavior
Activity 5.1a: Spinning Pennies
Section 5.1 - Constructing Models of Random Behavior
The Law of Large Numbers
In random sampling, the larger the sample, the closer the
proportion of successes in the sample tends to be to the
proportion in the population.
The difference between a sample proportion and the
population proportion must get smaller as the sample size
gets larger.
Section 5.1 - Constructing Models of Random Behavior
The Law of Large Numbers
Example: Fifty Fathoms demos Law of Large Numbers and
Law of Large Numbers 2
Section 5.1 - Constructing Models of Random Behavior
The Fundamental Principle of Counting
For a two-stage process with n1 possible outcomes for stage
1 and n2 possible outcomes for stage 2, the number of
possible outcomes for the two stages taken together is n1n2.
More generally, if there are k stages, with ni possible
outcomes for stage i, then the number of possible outcomes
for all k stages taken together is n1n2 …nk.
Section 5.1 - Constructing Models of Random Behavior
Tree Diagrams
Example: A tree diagram of all possible outcomes when flipping a fair
coin twice.
Flip 1
Flip 2
Outcome
H
HH
T
HT
H
TH
T
TT
H
T
Section 5.1 - Constructing Models of Random Behavior
Two-way Tables
Example: There are 36 equally likely outcomes when rolling two dice.
Second Roll
First
Roll
1
2
3
4
5
6
1
1,1
1,2
1,3
1,4
1,5
1,6
2
2,1
2,2
2,3
2,4
2,5
2,6
3
3,1
3,2
3,3
3,4
5,3
3,6
4
4,1
4,2
4,3
4,4
5,4
4,6
5
5,1
5,2
5,3
5,4
5,5
5,6
6
6,1
6,2
6,3
6,4
6,5
6,6
Section 5.1 - Constructing Models of Random Behavior
Summary
A probability model is a sample space together with an
assignment of probabilities.
The sample space is a complete list of disjoint outcomes
where
• Each outcome is assigned a probability between 0 and 1
• The sum of all the probabilities is 1.
S  {a1 , a2 ,..., an }
0  P(ai )  1
n
 P(a )  1
i
i1
Section 5.1 - Constructing Models of Random Behavior
Summary
The probability of an event is the number of outcomes that
make up the event divided by the total number of possible
outcomes.
Section 5.1 - Constructing Models of Random Behavior
Summary
The main practical application of equally likely outcomes are
in the study of random samples and in randomized
experiments.
In a survey, all possible simple random samples are equally
likely.
In a completely randomized experiment, all possible
assignments of treatments to units are equally likely.
Section 5.1 - Constructing Models of Random Behavior
Summary
The only way to decide whether a probability model is a
reasonable fit to a real situation is to compare probabilities
derived from the model with probabilities estimated from
observed data.
Section 5.1 - Constructing Models of Random Behavior
Summary
The Fundamental Principle of Counting
If you have a process consisting of k stages with ni outcomes
for stage i, the number of outcomes for all k stages taken
together is n1 n2 n3 · · · nk