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**Probability and the Binomial**

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

Cards. Weather. Other Examples? Definition (For use in this class). For a situation in which several different outcomes are possible, the probability for any specific outcome is defined as a fraction or proportion of all the possible outcomes. Probability of A. p(A) = (number of outcomes classified as A)/(total number of possible outcomes).

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**Tossing a Coin What are the possibilities?**

Heads Tails What is the probability of tossing a head? There is one head There were two possibilities Therefore, one in two

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**What Is The Range of Probabilities?**

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

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

No, then why do I care about probability? We use concepts from probability to determine the likelihood of choosing certain scores, or groups of scores (samples), from a population distribution

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**A Simple Demonstration**

We have a set of scores {1,1,2,3,3,4,4,4,5,6} What is the probability that we choose a number greater than 4? p(X>4)= 2/10 = .20 = 20% If you are unsure of this math, please review appendix a

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

In the previous example we had n = 10 What happens to the distribution when n = a very large number? The distribution becomes a smooth curve Most of the time the distribution becomes normal

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

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**How Can We Look at Score Probabilities Based on The Normal Distribution?**

First we must convert raw scores into z-scores From here, based on the normal curve, we can use a chart to determine probabilities

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**What Is the Unit Normal Table?**

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

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

B + C = 1.00 D + C = 0.50

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Lets Try A Few Examples

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

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**What Is The Binomial Distribution?**

The Binomial distribution is used when two categories exist naturally in the data For example, heads or tails on a coin In the case of heads and tails: p(heads) = p(tails) = ½ We will usually have questions such as: What is the probability of obtaining 15 heads in 20 tosses of a fair coin? The normal distribution does an excellent job of answering these questions

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**Notation and Assumptions**

Two categories, A and B p = p(A) = the probability of A q = p(B) = the probability of B The variable X refers to the number of times category A occurs in the sample

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**More About the Binomial Distribution**

Therefore, the binomial distribution shows the probability associated with each value of X from X = 0 to X = n. What does X = 0 mean? There are no instances of A in the sample (therefore it is all B) What does X = n mean? There are ONLY A’s in our sample, and therefore no B’s

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**The Binomial Is Eventually Approximately Normal**

When pn and qn are both equal to or greater than 10 When this happens: Mean: μ = pn Standard deviation: σ = √(npq) We find z-scores by: z = (X – pn) / √(npq) Remember z = (X – μ) / σ

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Section 7.2. Mean of a probability distribution is the long- run average outcome, µ, or µ x. Also called the expected value of x, or E(X). µ x = x i P.

Section 7.2. Mean of a probability distribution is the long- run average outcome, µ, or µ x. Also called the expected value of x, or E(X). µ x = x i P.

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