# Probability and the Binomial

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Probability and the Binomial

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).

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

What Is The Range of Probabilities?
0 – 1 What does a probability of zero mean? What does a probability of one mean?

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

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

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

Once Again Ladies and Gentleman: The Normal Distribution

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

What Is the Unit Normal Table?
It is a table that gives us proportions of scores in a normal distribution based on z-scores

What Are Some Relationships We Notice From the Chart?
B + C = 1.00 D + C = 0.50

Lets Try A Few Examples

What Is Special About z = 1.96?

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

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