AP Statistics Unit 4: Probability

Probability

How likely an event is to occur, the long run relative frequency of an event. Cannot predict specific outcome for a random event, only what is going to happen in the long run. Notation: P( ). Written as a decimal

And, Or, Not

And: P(AnB)=P(A)xP(B), must be independent (multiplication rule)

Or: P(AUB)=P(A)+P(B), must be disjointed

Not: 1-P(A)

Independence

Outcome of one trial does not influence the outcome of another. To assume independence, sample must be random. P(A/B)=P(A)

Law of Large Numbers

Long run relative frequencies of repeated independent events settle down to the true probability as the number of trials increases.

Law of Averages

An outcome is due to happen if it has not happened in a while. Is false, because each event is independent.

Sample Space

Set of all possible outcomes. Sum of sample space is 1

Complement

Probability that the event does not occur.

Complement Rule: Probability of an event is 1-complement (probability that event does not occur).

"At least one"

Find probability of event not occurring and do 1-P(not happening).

Outcome

Result of a trial, which is measured, observed, or reported (ex. color of M&M pulled out of bag)

Event

Collection of outcomes. Have probabilities attached to them. (ex. pulling out 3 green M&Ms)

Disjoint/Mutually exclusive

Two events do not share outcomes. Therefore, if A occurs, B cannot occur. (ex. playing card cannot be hearts and spades, so they are disjoint)

Something has to happen rule

Sum of all probabilities of all possible outcomes is 1

Conditional Probability

Usually use contingency tables. Formula: P(A/B)=P(AnB)/P(B). aka overlap divided by "second thing" (general multiplication rule). We know something about the event.

Venn Diagram

Show two non-disjoint events, intersection of circles is the probability of both, rectangle enclosing box represents neither.

General Addition Formula

P(AUB)=P(A)+P(B)-P(AnB), P(AnB) is probability of both

Replacement

Returning sampled items back to the population. Samples usually done without replacement (so items are NOT returned). When drawing from small populations without replacement, probabilities must be altered for each instance (If I draw 3 cards...).

Tree Diagram

A display of conditional probabilities (sequence of probabilities). Multiply across the branches to find probability of two events

Expected Value

E(x)=sum of x*P(x), the average/mean outcome after repeated events. The center (mu). "Based on the probability model,_____ would expect to _____ an average of ___E(x)____ for each time ______ happens" (can only be found if random variable is discrete)

Standard Deviation

Used to combine random variables. Used with normal model. Generally, it informs us about how the outcomes vary (spread). Variance (var(x)) is S(x)^2

Random Variable

Value based on a random event (ex. tossing a coin, head or tails is random event)

Shifting/Rescaling

Shifting: add a constant to every number, E(x) is affected

Rescaling: Multiply a constant to every number, E(x) and S(x) affected

Discrete Random Variables

Can list all possible outcomes (in a probability model)

Continous Random Variables

Can take any value, within a range of numbers. Range can be infinite or bounded

Combining Random Variables

E(x): just add (or subtract) together

S(x): S(x)^2+S(x)^2=S(x)^2 (Pythogrean theorem of stats)

Variances always add

Probability Model

Function that associates a probability, P, with each value of a discrete random variable, X, denoted P(X=x). Can also assign probailities to ranges of continuous random variables.

Bernoulli Trials: 3 conditions

1. Define two outcomes, success (p) and failure (q, or 1-p)

2. Probability of p stays the same for every trial

3. Trials are independent (10% condition: sample is smaller than 10% of population)

Geometric Probability

Trying to find first success. Expected number of trials until first success is E(x)=1/P(success). Counts number of Bernoulli trials until first success.

Binomial Probability

Counts number of success (X or k equals number of successes) with fixed number of Bernoulli Trials (n=number of trials). binompdf( is used for "exact", binomcdf( is used for "at least" or "between" or "at most".

Success/Failure Condition

If np>10 and nq>10, then we can use normal model for binomial probability. (must expect at least 10 successes and 10 failures) mean=np, s(x)=(npq)^1/2

Binomcdf and binompdf

binompdf= gives probability of X successes given number of trials (n)

binomcdf= gives probability of X successes or less (to left) given n trials. (at least is 1-binomcdf)

Mutually exclusive

events that cannot occur at the same time.

Outcomes in collection

all possible outcomes- even if they duplicate- for example, 1 and 4, and 4 and 1 would be two different outcomes

Probability is in the _______

Long run

“and symbol” probability

“Or” symbol

Event

a set of outcomes in a probability experiment, representing a specific result or combination of results.

if the probability of two things happening is 0, the events are:

mutually exclusive

random variable

In AP Statistics, a random variable is a variable whose value is determined by the outcome of a random phenomenon, and its possible values and probabilities are described by a probability distribution.

Mean of random variable distribution (X+Y)

just add them

How to tell if table is indepdent

x and y increase in constant patterns +2, +3, etc

standard deviation represents:

POPUlation

How are the mean and SD effect when multiplied by a constant

Mean: Multiplied

SD: Multiplied

How are the mean and SD affect when you add or subtract by a constant

Mean: shifted

SD: unchanged