Thinking About Chance &

Probability ModelsChapters 17 and 18

November 14, 2012

Chance Processes

Randomness and Probability

Probability Models

Probability Rules

Using Probability Models for Inference

How Probability Works

1.0 Chance Processes

People talk loosely about chance all the time.

For scientific purposes, chance has a precise meaning.

Chance behavior is unpredictable in the shortrun but has a regular and predictable pattern inthe long run.

This definition pre-supposes that the process can berepeated over and over again, independently and underthe same conditions.

Many games fall into this category.

1.1 Heads or tails?

One simple game of chance involves betting on the tossof a coin.

The result cannot be predicted in advance.

But there is still a regular pattern in the results:proportion of heads seen in the long run.

The following figure shows the results of tossing a coin1,000 times: HTTHHHT . . .

1.2 Random Sampling

Choose a S.R.S. of size n from a population.

Suppose the unknown parameter is p.

Then p̂ converges to p for large n. (Law of largenumbers)

Then the distribution of the sample estimate p̂converges to a normal density curve in the long run.

p

2.0 Randomness and Probability

A variable is called a random variable if its values areuncertain but there is some long run stability in theprocess underlying the data.

protagonist helper hinderer

Variable?

Observational units?

2.1 Example

Variable X: “Whether an infant chose the helper orhinderer”.

Each infant is like one repetition of the experiment,presumably under identical conditions.

If we assume that each infant is making the planned useof chance in their selection, then X is a random variable.This is called choosing at random.

Then the proportion of infants who choose the helper(say) will stabilize in the long run.

This long run stability point is then called theprobability of choosing the helper.

3.0 Probability Models

A probability model for a random variable describesthe possible outcomes and says how to assignprobabilities to them.

Random variable X: Kind of toy selected.

Table: Probability Model for X

X ProbabilityHelper p

Hinderer (1-p)

3.1 Probability Model for S.R.S

Suppose we take a S.R.S. of size n from a population.Let p be the unknown proportion of individuals from thepopulation who share a certain characteristic.

Let p̂ be the proportion of individuals in our sample whoshare this characteristic.

Then the Normal density curve assigns probabilities to thepossible values of p̂ upon repeated sampling so long as nis large. The mean of this density is p. The standard

deviation is√

p(1−p)n

.

4.0 Some Probability Rules

A Any probability is a number between 0 and 1.

For any outcome A, the probability P(A) satisfies0 ≤ P(A) ≤ 1.

B All possible outcomes together must have probability 1.

C The probability that an outcome does not occur is 1minus the probability that it does occur.

D The probability that one or the other outcome occursis the sum of their individual probabilities.

4.1 Example

Ex 18.8: Select a first year college student at random and askwhat their academic rank was in high school. Here is theprobability model for the random variable X = academic rankof student.

Rank Top20%

Second20%

Third20%

Fourth20%

Lowest20%

Prob. 0.44 0.26 0.23 0.06 0.01

1. Check that this is valid probability model.

2. What is the probability that a randomly chosen first yearcollege student was not in the top 20%?

3. What is the probability that a randomly chosen first yearcollege student was in the top 40%?

4.2 More Probability Rules

Multiplication Rule: The chance that two things willboth happen equals the chance that the first will happen,multiplied by the chance that the second will happengiven the first has happened.

Two things are independent if the chancesfor the second given the first are the same, nomatter how the first turns out. Otherwise theyare dependent.

Addition Rule: To find the chance that at least one oftwo things will happen, check to see if they are mutuallyexclusive. If they are, add the chances.

Two things are mutually exclusive whenthe occurrence of one prevents the occurrenceof the other; one excludes the other.

5.0 Lotsa Distributions

(from Prof. June Morita)

= sampling unit

POPULATION SAMPLEdesired information approximation/estimate

parameter

numerical fact re. population

statistic

estimates the parameter

We want to estimate the proportion of individuals in apopulation with a certain characteristic – call it p.Draw an S.R.S. of size n individuals from the population.Compute the sample proportion p̂. Use it to estimate p.

5.1 Population Distribution

DefinitionThe population distribution of a variable is thedistribution of the values of the variable among all individualsin the population.

Example: Elway Research conducted a poll of 408 randomlyselected Washingtonians and found that 64% (or 261) of theirsample said they would support a temporary sales tax hike.Write the population distribution for the variable: would yousupport a sales tax hike? (yes/no).

(All possible values for the variable and % of time it occurs inthe population.)

5.2 Distribution of Sample Values

DefinitionThe distribution of the values in the sample showsall possible values seen for the variable in our sample and howoften those values occured.

Example: Elway Research conducted a poll of 408 randomlyselected Washingtonians and found that 64% (or 261) of theirsample said they would support a temporary sales tax hike.Write the distribution of the sample values for the variable:would you support a sales tax hike? (yes/no).

5.3 Sampling Distribution of Statistic

DefinitionThe sampling distribution of a statistic shows thedistribution of values of the statistic in all possible samples ofsize n drawn from the population.

Example: Elway Research conducted a poll of 408 randomlyselected Washingtonians and found that 64% (or 261) of theirsample said they would support a temporary sales tax hike.Write the sampling distribution of their statistic p̂?

For large sample sizes, p̂ tends to be close to theparameter p. (law of large numbers)

And as the sample size increases, the samplingdistribution for p̂ looks more and more like a normaldistribution. The mean of this density is p. The standard

deviation is√

p(1−p)n

. (central limit theorem)

5.4 Compare and Contrast

The population distribution describes the individualsthat make up the population.

versus

The distribution of the values in the sampledescribes the individuals that make up the sample;

versus

The sampling distribution describes how a statisticvaries from one sample to another, all drawn from the samepopulation.

5.4 Example

You ask an S.R.S of 1,500 college students whether theyapplied for admission to any other college. Suppose 500 (33%)students in your sample admitted they had.

1. Population? Sample? Sample Size?

2. Variable?

3. Population distribution of variable?

4. Distribution of variable in sample?

5. What is the statistic (in words)? What is its’ value forthis one sample?

6. What is the sampling distribution of this statistic?

6.0 Sampling Distribution As a

Probability Model

Sampling Distributions Are Probability Models.

What does this mean exactly? It meansI Imagine having a box full of tickets. Each ticket has a

value of p̂ on it.I How many there are of each p̂ will depend on the

sampling distribution.I We mix up these tickets well and draw one at random.I Then the probability of picking a ticket with a specific p̂

value will be determined by the area given to that valueof p̂ under the sampling distribution.

Check that the sampling distribution for p̂ from example5.4 is a valid probability model.

6.1 Example

You ask an S.R.S. of 1,500 college students whether theyapplied for admission to any other college. Suppose, in fact,35% of all college students applied to colleges besides theones they are attending.

1 What is the sampling distribution for p̂?

2 What is the probability that if you select at random fromthis distribution, you will get a p̂ less than 0.33?

3 What is the probability that if you select at random fromthis distribution, you will get a p̂ less than 0.33 or greaterthan 0.35?

4 Give the range of the middle 95% of the p̂ values for thisdistribution. Interpret this range.

7.0 How Probability Works

The probability of a head in tossing a fair coin is 0.5.

This means that as we make more tosses, theproportion of heads will eventually get close to 0.5.

This does not mean that the count of heads will getclose to half the number of tosses. Why?

Number of Proportion of Number ofof Tosses of heads heads

100 0.51 511,000 0.507 507

10,000 0.5031 5031100,000 0.50074 50,704