Stats 2nd test

classical probability

classical probability

uses theory to calculate probability (math)

empirical probability

only uses observation to calculate probabilities (stats)

Subjective Probability

uses personal knowledge to create likelihoods (decision making ability)

The Law of Large Numbers

as experiments are repeated without bound the observed probability will approach the theoretical probability

probability

science of attaching likelihoods to random occurrences

experiment

any process with the likelihood of random outcomes

sample space

the set of all possible outcomes of an experiment

the event

subset of the sample space, what we want to see/successful outcomes

Basic probability formula

probability of event = #outcomes in event/#outcomes in sample space (P(E)=E/S)

mutually exclusive

when 2 events have no outcomes in common

basic addition rule

if A&B are mutually exclusive then probability of A OR B is P(A) + P(B)

the addition rule

when not mutually exclusive P(A&B) = P(A)+ P(B) - P(A&B)

complements

an event E is the complement of E^c if they have no outcomes in common and E+E^c equals the total sample space

the multiplication rule

for when events happen in succession:

events must be independent

P(A&B)= P(A)* P(B) “and then”

conditional probability

P(A|B) “A given B”

this uses a given condition that causes events to have a dependency

geometric probability

used for probabilities looking for the first success

formula: P(nth) = pq ^n-1

p-success

q-failure

binomial probability

-need a set # of trials

-only two outcomes (success/failure)

-all trials are independent

P( exactly A) =p^A * q^Ac * nCr

p-probability success; q-probability failure; Ac- complement of outcomes/ n-A; nCr-combination

probability distributions

a table that represents the outcomes of an experiment with the given probabilities

x | P(x) | xP(x)

the mean of xP(x) is Expected Value

EV = sum of xP(x)

Normal Distribution

• mean falls in the middle

• graph is symmetrical

• graph floats above x axis

• area under whole graph = 1

bell shaped curve

Standard Normal Distribution

• zero fall in the middle

• measure by std dev

• graph is symmetrical

• graph floats above x axis

• area under curve = 1

z-scores

measure the distance from the mean using std devs

z = (x-mean)/std dev

percentiles

always lean left tail/shading

75th percentile means less than 75%

central limit theorem

when data ISNT normal

as the size of samples taken from a population increases without bound the graph of the sample means will approach normal and the mean of the sample means will equal the population mean and the std dev of the sample means will equal the population std dev divided by the square root of the sample size

n>30 assume normal

mew of x-bar = mew

std dev of x-bar = std dev/root(n)

new z-score formula

z = (xbar-mew)/(sigma/root(n))

proportions

p-hat - sample proportion for success

w-hat - sample proportion for failure

central limit theorem for proportions

if nqp > 10 then assume normal

z = (phat-p) / root(pq/n)

binomial probability

“at most” “at least”

use binomial calculator