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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
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
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