Probability Unit 4 - Statistics

What is probability? Proportion of times any outcome of any random phenomenon would occur in a very long series of repetitions Long-term relative frequency (probability settles down) Branch of math that describes the pattern of chance outcomes Not a guarantee!

Is a number between 0 and 1. All possible outcomes together must total 1. Complement Rule - The probability that an event does not occur is 1 minus the probability that it does. Probability Rules

Probability in Two-Way Tables Classifies each person by 2 variables (row and column) Marginal Distributions –Found at the bottom or right margin –Are entire rows/columns over the total Conditional Distributions –Only a cell that satisfies a certain condition (given in the row/column)

Segmented Bar Graphs Segments correspond to conditional probabilities but every bar totals 100% Use different shading and include a legend!

Simpson’s Paradox The reversal of the direction of a comparison when switching from conditional to marginal –Hospital A vs. Hospital B

Independent Events Two categorical variables are independent if the conditional distributions of one are the same for every category of the other Math Rule: If p(A) = p(A|B), then A and B are independent events

Combinations and Permutations Combinations –Order doesn’t matter Permutations –Order does matter

Binomial Setting Each observation falls into one of just two categories, “success” or “failure” There is a fixed number n of observations The n observations are all independent The probability of success, p, is the same for each observation

Binomial Probability Formula

Simulations Estimate probabilities we don’t know or confirm ones that we do Calculator –randInt(lowest, highest, how many) –randBin(total, probability, number of times) Fathom –randomPick –randomInteger Dice, Coins, etc

Expected Value Long-Term Outcome (mean) Fair Games Formula:

Sampling Distributions and Sample Size Statistics vary naturally from sample to sample The larger the sample size, the less sampling variability Population size does not effect the sampling variability

Law of Large Numbers For any distribution, as the number of observations increases, the overall mean or proportion will approach the true value How large is large enough?