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

Probability-The 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 DiagramsThe 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

Population-

The 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

Sample- A 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

Random-An 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 Distribution-Is 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 Distribution-We 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

Independence-Two 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

Dependent-When 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 re-select, 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

Replacement-Refers 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 Probability-The 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 Probability-The 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

BA and B

Probability

Intersection-

The probability that all events occur.

A

BA and B

Probability

Conditional Probability-

To 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 Probability-

The 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 Probability-

A 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 Probability-

Assume 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 Probability-

Now 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:

A

BA and B

AB A and B

Small Large

Joint Probability-

What 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 Probability-

To answer this we need to know if the outcomes are mutually exclusive—does the

occurrence of one event preclude the other from occurring?

Joint Probability-

Are 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 Probability-

If yes, then the events are dependent.

If No, then they are independent.

A

B

A

BA and B

Yes

‘Disjoint’

No

Joint Probability-If 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 Probability-

If 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 Probability-

To 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 Probability-

Example: 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?