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

!)!(

!

rrn

nCrn

)!(

!

rn

nPrn

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 rnr

rn ppC )1(

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: ( ) i iE x x p

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?