Transcript Probability - collingwoodresearch

POSC 202A: Lecture 4
Probability
Probability
We begin with the basics of probability and then move
on to expected value.
Understanding probability is important because the
rules that govern probabilistic relationships allow us
to explicitly account for the stochastic nature of the
world.
We begin with some basic definitions…..
Probability
ProbabilityThe percentage of times something is expected
to happen, when the process is infinitely
repeated, under the same conditions.
Probability
Since it is a percentage, it ranges from 0 (never
expected to occur) to 1 (it always occurs).
Thus we can calculate a probability that something
occurs by subtracting the probability that it does not
occur from 1.
Venn Diagrams
The area in the rectangle=1 which is the sample space.
The probability that event ‘A’ occurs is indicated by the
size of the area that is covered in the space.
The complement is the area in the sample space not
covered by the circle ‘A’. It is abbreviated Ac.
So: Ac= 1-A
A
Ac
Probability
Thus we can calculate a probability that
something occurs by subtracting the
probability that it does not occur from 1.
Example:
If a coin is 2 sided, the probability that heads will
occur is 1- (the probability of tails).
Probability
PopulationThe entire group of things about which we want
information.
An estimate that describes an aspect of the
population is a parameter. A parameter is a
number that describes the population – in
theory it is fixed, but rarely do we know it.
An example: The US Census.
Probability
SampleA part of the population from which we actually
collect information, used to draw conclusions
about the population.
An estimate that describes an aspect of the
sample (our quantity of interest) is a a
statistic.
Probability
RandomAn event is random when its selection can be
attributed to chance.
We think of random selection occurring when all
units have an equal chance of being
selected.
Example: roll of dice.
Probability: a brief tangent
Sampling DistributionIs the distribution of the values of some statistic in all
possible samples (of a particular size) that could be
taken from a population.
This is a distribution in which each observation or unit is
itself the product of a sample.
The sampling distribution is Normal about the true
population mean.
Underlying distributions converge around the sample
mean as the number of trials increase.
Estimating the mean of a
distribution (or variable)?
We can think of two ways to do this:
1. Calculate the average.
-(Not always possible)
2. Simulate through repeat sampling.
-In essence, we estimate the sampling distribution.
Probability: a brief tangent
Class exercise.
Pair up. We are going to create a sampling distribution.
1. Roll the dice. (in R, hint: requires sample() function)
2. Record each roll. (Repeat 5 more times-6 total)
3. Calculate and record the average of the 6 rolls.
4. Repeat 20 times. You should have 20 averages.
Each average represents the result from a sample.
Probability: a brief tangent
Sampling DistributionWe can create a sampling distribution of the means of
all groups samples.
What do we find when we graph the average of the
samples?
Why is this useful?
Probability
IndependenceTwo things are considered independent if the
chances of the second occurring, given our
knowledge of the first, are the same.
The selection of one does not effect the chances
that another is selected.
Example: the second roll of the dice is unrelated
to the first.
Probability
Example: the second roll of the dice is unrelated
to the first. So the probability of any number
remains 1/6.
A common misperception is that the number that
just came up is less likely to occur on the
next roll.
If that were true, what would we expect to see in
our results?
Probability
DependentWhen the outcome of future events change due
to events that precede them.
The selection of one DOES effect the chance
that another is selected.
Dependent- vs. Independent
Example: Pick a card from a deck (of 52).
What is the probability of selecting any one card?
If we place the card back in the deck and reselect, then the second selection is
independent of the first.
If we do not place the card back in the deck then
the chances for selecting a particular card are
dependent on the selection of the first card
(1/51).
Probability
ReplacementRefers to returning the item selected to the
sample so that the odds of selecting any one
case remain unchanged (the draws remain
independent).
Probability
Two general types of more complex processes.
Conditional probability:
What is the likelihood of one thing and the other
happening.
Joint probability:
What is the probability of one thing or the other
happening?
Probability
Conditional ProbabilityThe chance an event occurs given another event
has already occurred.
Conditional probability deals with dependent
events (though the individual components of
each step may be independent).
(Think of it as a two-step process)
Probability
Conditional ProbabilityThe chance an event occurs given another event
has already occurred.
Here we are trying to find the area ‘A and B’
which is called the intersection
A
B
A and B
Probability
IntersectionThe probability that all events occur.
A
B
A and B
Probability
Conditional ProbabilityTo find a conditional probability, use the
multiplication rule:
CP=Pr(a)*(Pr(b)given (a)).
Example: What are the odds of selecting the 7 of
clubs and then the queen of hearts:
CP=1/52*1/51
Conditional ProbabilityThe individual events can be independent or
dependent. We can account for both.
Assume independence.
Example: What is the probability of flipping a coin and
getting heads twice?
Two heads=.5*.5
Conditional ProbabilityA more complex example:
Assume independence.
Example: What are the odds that we will draw a female
Republican from a sample if the probability of
drawing a female is .53 and of a Republican is .5?
CP=.53*.5=.265
Conditional ProbabilityAssume dependence.
Now we need the probability of B given A.
In this case the proportion of Republicans among women.
Example: What are the odds that we will draw a female
Republican from a sample if the probability of
drawing a female is .52 and of a Republican is .4?
CP=.52*.4= ~.21
Conditional ProbabilityNow we need the probability of B given A.
One limitation is that when the probabilities are dependent, we
often don’t know what the probability of B given A is:
Small
Large
A
B
A and B
A
B
A and B
Joint ProbabilityWhat is the probability that more than one
outcome occurs (in a round)?
(e.g., what is the probability we roll a 1 or a 3?)
Joint ProbabilityTo answer this we need to know if the outcomes
are mutually exclusive—does the
occurrence of one event preclude the other
from occurring?
Joint ProbabilityAre the outcomes mutually exclusive—does
the occurrence of one event preclude the
other from occurring?
If yes, then the events are dependent.
(one depends on the other NOT occurring)
If no, then the events are independent.
(they can occur simultaneously)
Joint ProbabilityIf yes, then the events are dependent.
If No, then they are independent.
A
A
B
B
Yes
‘Disjoint’
A and B
No
Joint ProbabilityIf yes, then the events are disjoint.
To solve: we add the probabilities.
Example: the probability of getting a king of
clubs or queen of hearts.
JP=1/52 + 1/52
Example: the probability of getting any king or
any queen.
JP=4/52 + 4/52
Joint ProbabilityIf no, then the events are independent.
(they can occur simultaneously)
To solve: we add the probabilities and then
subtract their intersection.
JP= (A+B)-(AxB)
Joint ProbabilityTo solve: we add the probabilities and then
subtract their intersection.
JP= (A+B)-(AxB)
Example: the probability of drawing any club or
any queen.
JP=(13/52+4/52)-(13/52*4/52)
JP=(.25+.077)-(.25*.077)
Joint ProbabilityExample: the probability of drawing a Female or
a Republican in a sample.
Females are 52% of the population
Republicans are 40% of the population.
JP=(.52+.40)-(.52*40)
=(.92-.21)
=71%
Probability: Quick Summary
Two general types of more complex processes.
Conditional probability:
What is the likelihood of one thing and the other
happening.
Joint probability:
What is the probability of one thing or the other
happening?