Probability the likelihood of specific events

Probability We are going to focus on three types of probability problems Normal Curve Probabilities Binomial situations Simple combined event probabilities

The Normal Curve The Natural Law of Errors Properties of the Normal Distribution Percentile Ranks based on the Normal Distribution

The Natural Law of Errors Many things in nature are distributed normally Heights and weights, as well as other physical features of many living things IQ scores and many psychological and educational variables and attributes A form of order in the universe – p. 91 in book There is a typical, or average value, and there is just as much deviation above as below that middle point in a specific symmetrical pattern.

Properties of the Normal Dist. Unimodel Symmetrical Mean, Median, Mode – All at same place Bell shaped

Properties of the Normal Dist.

Properties of the Normal Dist. Has a particular mathematical function Infinite family of distributions Goes asymptotic to X axis Inflection points – 1 SD above and below mean

Properties of the Normal Dist.

Properties of the Normal Dist. Areas under portions of the curve are well tabulated – See pp. 448-451 in book All normal distributions can be converted to the Standard Normal Distribution z = ( x – m ) / s Empirical Rule +/- 1 SD ≈ middle 68%, +/- 2 SDs ≈ middle 95% +/- 3 SDs ≈ middle 99%,

Properties of the Normal Dist.

Percentile Ranks Interactive View of the Normal Distribution Class Exercises

Binomial Distribution We can use the known properties of the Binomial distribution to calculate the probability of specific kinds of events. The conditions that must be met: The event has only two outcomes (success / failure) across a fixed number of trials The probability of success is known and the same for each trial The events are independent, no event influences future events

Sample Binomial Situations Flipping a fair coin (head / tail) Having a baby (boy / girl) Selecting a card from a fair deck ( red / black) Shooting a foul shot in basketball Sampling a product from an assembly line and getting a defective product or a properly made product

Binomial Situations Simulation #1 Simulation #2 Class Exercises

Combined Event Probabilities What is the probability, using two fair dice, of rolling a 4 and a 6? What is the probability, using two fair dice, of rolling a 4 with one die or a 6 with the other die? What is the probability of rolling a 4 or a 6 with one roll? What is the probability, using two fair dice, of rolling two 6s? Calculations