PHA Test on Probability and Statistics

inclusion-exclusion principle

A u B = A + B - (A n B)

mutually exclusive

two things that cannot happen at once

not mutually exclusive

two things that can happen at once

multiplication principle

If an action can be performed in n1 ways and for each of these ways another action can be performed in n2 ways, then the two actions together can be performed in n1\*n2 ways.

addition principle

If two actions are mutually exclusive, and the first can be done in n1 ways and the second in n2 ways, then one action or the other can be done in n1 + n2 ways.

complement principle

if A is a subset of a universal set U, then:

A = U - complement of A

permutation

number of ways of getting some results in a *specific order*

n!/(n-r!)

combination

number of ways of getting some results without caring about order

n!/(n-r)!r!

number of permutations of things not all different

n!/(n1!*n2!*n3!\*n4!…)

n1 = number of type 1 elements

n2 = number of type 2 elements

etc…

probability of **either** of two events A and B if not mutually exclusive

P(A or B) = P(A) + P(B) - P(A n B)

probability of **either** of two mutually exclusive events A and B if mutually exclusive

P (A or B) = P(A) + P(B)

probability of two events A and B occurring together

if not independent:

P(A and B) = P(A) \* P(B given A)

\
if independent:

P(A and B) = P(A) \* P(B)

binomial probability theorem

k successes from n trials

p probability of individual k

n!/(n-k)!k! \* *p^k* \* (1-p)^n-k

permutation on calculator

nPr(n total things, r results)

order matters

combinations on calculator

nCr(n total things, r results)

order doesn’t matter

binomial probability of exact number of successes

binomPdf(n trials, probability of individual success, x successes)

binomial probability of at least a certain number of successes or a range of successes

binomCdf(number of trials, prob of success, lower bound, upper bound)

descriptive statistics

organization of data

inferential statistics

making inferences based on data

stem-and-leaf plot

first digit of each score as stem, leaves are rows from those digits

histogram

one axis as range of data, other as frequency

frequency table

one column for range of data

other column as frequency

other column as cumulative frequency (sum of the previous frequencies)

mean

average

median

middle number if set written out (if odd)

if even it is average of the middle two

mode

data point that occurs most often in set

box and whisker point

shows lower extreme (farthest left), 1st quartile (median of the data to the left of the median), median, 3rd quartile (median of the data to the right of the median), and upper extreme)

range = distance between two extremes

Bayes’ Theorem

P(A given B) = (P(A) \* P(B given A))/P(B)

expected value

probability of win \* *what you win + probability lose ** what you lose

variance

average of the squares of the deviations from the average of the entire set

\
s^2

(x1-mean of data)^2…/number of data points (n)

standard deviation

square root of variance

sample variance

average of the squares of the deviations from the average of the entire set

\
s^2

(x1-mean of data)^2…/number of data points minus 1 (n - 1)

sample standard deviation

square root of variance

sample vs not sample

sample has only portion of total

not sample is all data available

ie 10 random ppl vs a census

3 properties of normal curve

about 68% of the distribution is within 1 standard deviation of the mean

About 95% of the distribution is within 2 standard deviations of the mean

About 99% is within 3 standard deviations of the mean

standard normal distribution

normal distribution with mean of 0 and standard deviation of 1

total area under curve = 1

area of the curve to the left of a number is the proportion of the data that has a standard value less than z

using standard normal distribution table

z score (from table) = raw score ( number being looked at) - mean of data / standard deviation of data

\
take percentile you are given and look for closest z value then plug it in to equation

\
if given raw score solve for z

finding raw score on calculator

invNorm(percentile, mean, standard deviation)

finding z score on calculator

invNorm(percentile, 0, 1)

finding percentile on calculator (given raw score)

normCdf(lower bound, z score calculated, 0, 1)

nonprobability sampling

not based on random processes

probabilty sampling

based on random processes

P(AnB) =

P(A) \* P(BgivenA)