Probability and the Binomial. What Does Probability Mean, and Where Do We Use It? Cards. Cards. Weather. Weather. Other Examples? Other Examples? Definition.

Probability and the Probability and the BinomialBinomial

What Does Probability Mean, What Does Probability Mean, and Where Do We Use It?and Where Do We Use It?

Cards.Cards. Weather.Weather. Other Examples?Other Examples? Definition (For use in this class).Definition (For use in this class).

For a situation in which several different outcomes are For a situation in which several different outcomes are possible, the probability for any specific outcome is possible, the probability for any specific outcome is defined as a fraction or proportion of all the possible defined as a fraction or proportion of all the possible outcomes.outcomes.

Probability of A.Probability of A. p(A) = (number of outcomes classified as A)/(total p(A) = (number of outcomes classified as A)/(total

number of possible outcomes).number of possible outcomes).

Tossing a CoinTossing a Coin

What are the possibilities?What are the possibilities? HeadsHeads TailsTails

What is the probability of tossing a head?What is the probability of tossing a head? There is one headThere is one head There were two possibilitiesThere were two possibilities Therefore, one in twoTherefore, one in two

What Is The Range of What Is The Range of Probabilities?Probabilities?

0 – 10 – 1 What does a probability of zero mean?What does a probability of zero mean? What does a probability of one mean?What does a probability of one mean?

Are We In This Class To Are We In This Class To Become Better Poker Players?Become Better Poker Players?

No, then why do I care about probability?No, then why do I care about probability? We use concepts from probability to We use concepts from probability to

determine the likelihood of choosing determine the likelihood of choosing certain scores, or groups of scores certain scores, or groups of scores (samples), from a population distribution(samples), from a population distribution

A Simple DemonstrationA Simple Demonstration

We have a set of scores We have a set of scores {1,1,2,3,3,4,4,4,5,6}{1,1,2,3,3,4,4,4,5,6}

What is the probability that we choose a What is the probability that we choose a number greater than 4?number greater than 4? p(X>4)=p(X>4)=

• 2/10 = .20 = 20%2/10 = .20 = 20%• If you are unsure of this math, please review If you are unsure of this math, please review

appendix aappendix a

What Happens When We Have What Happens When We Have More Scores?More Scores?

In the previous example we had n = 10In the previous example we had n = 10 What happens to the distribution when n = What happens to the distribution when n =

a very large number?a very large number? The distribution becomes a smooth curveThe distribution becomes a smooth curve Most of the time the distribution becomes Most of the time the distribution becomes

normalnormal

Once Again Ladies and Once Again Ladies and Gentleman: The Normal Gentleman: The Normal

DistributionDistribution

How Can We Look at Score How Can We Look at Score Probabilities Based on The Probabilities Based on The

Normal Distribution?Normal Distribution?

First we must convert raw scores into z-First we must convert raw scores into z-scoresscores

From here, based on the normal curve, we From here, based on the normal curve, we can use a chart to determine probabilitiescan use a chart to determine probabilities

What Is the Unit Normal Table?What Is the Unit Normal Table?

It is a table that gives us proportions of It is a table that gives us proportions of scores in a normal distribution based on z-scores in a normal distribution based on z-scoresscores

What Are Some Relationships What Are Some Relationships We Notice From the Chart?We Notice From the Chart?

B + C = 1.00B + C = 1.00 D + C = 0.50D + C = 0.50

Lets Try A Few ExamplesLets Try A Few Examples

What Is Special About z = 1.96?What Is Special About z = 1.96?

What Is The Binomial What Is The Binomial Distribution?Distribution?

The Binomial distribution is used when two The Binomial distribution is used when two categories exist naturally in the datacategories exist naturally in the data For example, heads or tails on a coinFor example, heads or tails on a coin

In the case of heads and tails:In the case of heads and tails: p(heads) = p(tails) = ½p(heads) = p(tails) = ½

We will usually have questions such as:We will usually have questions such as: What is the probability of obtaining 15 heads in 20 What is the probability of obtaining 15 heads in 20

tosses of a fair coin?tosses of a fair coin? The normal distribution does an excellent job of The normal distribution does an excellent job of

answering these questionsanswering these questions

Notation and AssumptionsNotation and Assumptions

Two categories, A and BTwo categories, A and B p = p(A) = the probability of Ap = p(A) = the probability of A q = p(B) = the probability of Bq = p(B) = the probability of B The variable X refers to the number of The variable X refers to the number of

times category A occurs in the sampletimes category A occurs in the sample

More About the Binomial More About the Binomial DistributionDistribution

Therefore, the binomial distribution shows Therefore, the binomial distribution shows the probability associated with each value the probability associated with each value of X from X = 0 to X = n.of X from X = 0 to X = n.

What does X = 0 mean?What does X = 0 mean? There are no instances of A in the sample There are no instances of A in the sample

(therefore it is all B)(therefore it is all B) What does X = n mean?What does X = n mean?

There are ONLY A’s in our sample, and There are ONLY A’s in our sample, and therefore no B’stherefore no B’s

The Binomial Is Eventually The Binomial Is Eventually Approximately NormalApproximately Normal

When pn and qn are both equal to or When pn and qn are both equal to or greater than 10greater than 10

When this happens:When this happens: Mean: Mean: μ = pnμ = pn Standard deviation: σ = √(npq)Standard deviation: σ = √(npq) We find z-scores by:We find z-scores by:

• z = (X – pn) / √(npq)z = (X – pn) / √(npq)• Remember z = (X – μ) / σRemember z = (X – μ) / σ