AP stats units 5-6

A∩B

"A" intersects "B"

The effect of adding/subtracting a constant (b) on a probability distribution

-adds to/subtracts from measures of center and location (mean, median, quartiles, percentiles) -Does not change measures of variability (range, IRQ, standard deviation) - does not change the shape of the probability distribution

the effect of multiplying/dividing by a constant (b) on a probability distribution

-multiplies/divides measures of center and location (mean, median, quartiles, percentiles) by b -multiplies/divides measures of variability (range,IRQ, standard deviation) by b -does not change the shape of distribution

calculating binomial probability with a graphing calculator

2nd, vars, binompdf, enter values,enter,

Basic Probability Rules

For any event A: 0≤P(A)≤1 If S is sample space in a probability model: P(S)=1 addition rule for mutually exclusive events: P(A or B) = P(A) + P(B) Complement Rule: P(A^c)=1-P(A) probabilities of equal outcomes: P(A)=number of outcomes in event A/ total number of outcomes in sample space

Geometric probability formula

If X has geometric distribution with probability p of success on each trial, the possible values of X are 1,2,3..... if x is any one of these values, P(X=x)=(1-p)^x-1 ·p

independant events

If knowing whether or not one event has occurred does not change the probability that the other event will happen. They are independent if P(A|B)= P(A|B^c)= P(A) or simply They are independent if P(AUB)=P(A)xP(B)

the effect of a linear transformation on a random variable

If y=a+bX is a linear transformation of random variable X, -the probability distribution of Y has the same shape as the probability distribution of X if b>0 -μY=a+bμX (mean=μ) -σY=|b|σX (because b could be negative)

discrete random variables with graphing calculator

List: enter values into L1 and probability into L2 Graph: stat plot, plot 1, histogram, Xlist:L1, Freq:L2 Window: adjust with X for values and Y for probability Graph Calc: stat, calc, 1-var

graphing binomial distribution on graphing calculator

List: list possible values of random variable X into L1 Highlight L2, binompdf(n,p), enter, make histogram Remember to include 0 with the values if you keep getting an extra number in L2

addition rule for mutually exclusive events

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

Complement Rule

P(A^c)=1-P(A) P=probability A^c= complement of event A A=event A

Probability distribution for discrete random variable

Probability distribution for discrete random variable X lists the values xi and probabilities (pi) which must be between 0 and 1 and add to be 1 P(xi) or P(X=xi)

Intersection

The event "A and B" is called the intersection of events A and B. It consists of all outcomes that are common to both events, and is denoted A∩B

geometric distribution

The probability distribution of X with probability p of successes on any trial. Possible values of X are 1,2,3....

multiplication rule for independent events

The probability that both events A and B will occur P(A and B)=P(A∩B)= P(A)xP(B)

AUB

The union of A and B.

discrete random variable

X takes a fixed set of possible values with gaps between

event

any collection of outcomes from some random process

Geometric setting

arises when we perform independent trials of the same random process and record the number of trials it takes to get one success. On each trial, the probability p of success must be the same

binomial setting

arises when we perform n independent trials of the same random process and count the number of times that a particular outcome (called a "success") occurs: Four conditions for a binomial setting are (BINS) -Binary?Possible outcomes of each trial can be classified as success or failure -Independant? trials must independant (knowing outcome of one cannot predict outcome of another) -Number? number of trials n of the random process must be fixed in advance -Same probability? There is the same probability of success p on each trial

mean (expected value) of a discrete random variable

average value value over many, many trials of the same random process. Σxi⋅P(xi) AKA multiply each value of x by its probability and add the products

continuous random variable

can take any value in an interval on the number line

A|B

conditional probability, probability of A given that B has occurred Basically, the | means given that

probability model

description of some random process that consists of two parts: a list of possible outcomes and the probability for each outcome

median of probability distribution

find where the probability of .5 lies by reading and adding the probabilities left to right, choosing the number that lands on .5 or passes it

standard deviation of sum of independent random variables

for any 2 independent random variables X and Y, if S=X+Y, the variance of S is σ^2S= σ^2(of X)+σ^2(of Y) Standard deviation is the square root of variance σ^2S= √σ^2(of X)+σ^2(of Y)

mean (expected value) of difference of a random variables

for any 2 random variables X and Y, if D=X-Y, the mean of D is μD=μX-μY

mean(expected value) of sum of a random variables

for any 2 random variables X and Y, if S=X+Y, the mean of S is μ(of S)=μ(of X)+μ(of Y) Both the sum of means and the mean of the sum work.

General Multiplication Rule

for any random process, the probability that both event A and B occur can be found using P(A and B)=P(A∩B)= P(A)xP(B|A)

random process

generates outcomes that are determined purely by chance

geometric probability on a graphing calculator

geometpdf computes P(X=x) geometcdf computes P(X≤x) 2nd, vars, enter values, enter

probability distribution

gives a random variable's possible values and its probabilities

mutually exclusive (disjoint)

if 2 events have no outcome in common and therefore can never occur together P(A and B)=0

general addition rule

if A and B are any 2 events resulting from some random process P(A or B)=P(A)+P(B)-P(A and B) Aka P(AUB)=P(A)+P(B)-P(A∩B)

probabilities of equal outcomes

if all outcomes in a sample space are equally likely, the probability that event A occurs can be found using formula P(A)=number of outcomes in event A/ total number of outcomes in sample space

mean (expected value) of binomial random variable

if count X of sucesses has a binomial distribution of with number of trials n and probability of success p, the mean of X is μ(of X)=E(x)=np

standard deviation of a binomial random variable

if count X of sucesses has a binomial distribution of with number of trials n and probability of success p, the standard deviation of X is σ(of x)=√np(1-p)

independent random variables

if knowing the value of X does not help predict the value of Y. If knowing one variable does not change the probability distribution of the other variable

law of large numbers

if we observe more and more trials of any random process, then the proportion of times that a specific outcome occurs approaches its probability

simulation

imitates a random process in such a way that simulated outcomes are consistent with real-world outcomes

sample space

list of all possible outcomes

standard deviation of discrete random variables

measure how much the values of variables typically vary from the mean in many, many trials of the random process √Σ(xi-μx)^2⋅P(xi)

venn diagram

one or more circles surrounded by a rectangle, with each circle representing an event. The region inside the rectangle represents the sample space of the process

tree diagram

shows the sample space of a random process involving multiple stages. The probability of each outcome is shown on the corresponding branch of each tree. All probabilities after the first stage are conditional probabilities

Binomial probability formula

suppose that X is a binomial random variable with n trials and probability p of success on each trial. The probability of getting exactly X successes in n trials (x=0,1,2...,n) is P(X=x)=(n x)p^x(1-p)^n-x where (n x)=n!/x!(n-x)!

Large counts condition

suppose that a count X of successes has a binomial distribution with n trials and successes probability p. The large counts condition says that the probability of distribution X is approximately Normal if np≥10 and n(1-p)≥10 That is, the expected numbers of both successes and failures are both atleast 10

Random variable

takes numerical values that describe the outcomes of some random process

binomial random variable

the count of successes X in a binomial setting. The possible values of X are 1,2,....,n

Union

the event "A or B" is called the union of events A and B. It consists of all outcomes that are in event A or event B, or both, and is denoted A∪B

complement

the event that A does not occur

geometric random variable

the number of trials X that it takes to get a success in a geometric setting

binomial coefficient

the numbers of ways to arrange X successes among n trials (binomial coefficient)=n!/ x!(n-x)! for x=1,2,...,n where n! is given by n!=n(n-1)(n-2)⋅⋅⋅(3)(2)(1) and 0!=1 (the binomial coefficient is not a fraction)

Binomial Distribution

the probability distribution of X. Completely specified by 2 numbers: number of traits (n) of the random process and the probability (p) of success

how to find probability for a continuous random variable

the probability for any event containing a continuous random variable is the area under the density curve and directly above the values on the horizontal axis that makeup the event. Remember the area/probability=1, and area=basexheight . So probability=heightxthe value you want the probability of

conditional probability

the probability that 1 event happens given that another is known to have happened. The conditional probability that event A happens given that event B happened is P(A|B) P(A|B)= P(A and B)/P(B)= P(A∩B)/P(B)= P(Both events occur)/P(given B event occurs)

Probability

the proportion of times an outcome of a random process would occur in a long series of trials, represented by a number between 0 and 1. It is unpredictable in the short run but has a regular and predictable pattern in the long run

standard deviation of difference of independent random variables

the variance is σ^2D= σ^2(of X)+σ^2(of Y) the standard deviation σ^2D= √σ^2(of X)+σ^2(of Y) It is the variance of the sum, NOT the sum of the variances (we can't assume they are independent)

Binomial coefficient on graphing calculator

type the n, math, PRB, nCr, type the x, enter

linear combination of independent normal variables

we must know that the graph is normal/independent to know distributions

10% condition

when taking random sample of size n from the population of size N, we can treat individual observations as independent when performing calculations as long as n<0.10N

variance

Σ(xi-μx)^2⋅pi

mean(expected value) and standard deviation of geometric random variable

μ(of x)=E(X)=1/p σ(of x)=√1-p/ p