AP Stat Unit 4

Random

Generates outcomes purely by chance

Probability

A number between 0 and 1 that describes the proportion of times the number would occur in a very long series of trials

Not a simulation

Law of Large numbers

Says if we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value that is the true probablity

Simulation

The imitation of chance behavior based on a model that accurately reflects the situation

Preforming a simulation

1) Describe how to set up and use a random process to preform one trial of the simulation

2) Preform many trials (20 is good to get a estimate)

3) Use the results of your simulation to answer the question of interest

Preforming a simulation to estimate the probability of getting at least one run of 3 heads in a row in 10 flips of a fair coin→ correct simulation description

Assign the numbers 0 through 4 to “heads” and 5 through 9 to “tails” Read 10 one-digit numbers from a random digits table and count this simulation as a “success” if there is at least one run of 3 or more heads, repeat 500 times and divide the total number of “successes” by 500

A or B or AUB

Union of A and B→ will consist of all outcomes in events A, B or both

Event

The specific outcome or set of outcomes of a random phenomenon

ex: flip coin and #, roll dice and 2

Probability model

Lists all the possible outcomes of an events and their probabilities

Sample space (S)

The list of all possible outcomes

A and B

intersection of A and B→ consists of all outcomes common to both events

To find probability that an event occurs

P(A)= # of outcomes in event A/ total # of outcomes in event space

P(S)=1

The sum of probability for the sample space

Complement formula

P(A^ complement)=1-P(A)

The probability A does NOT occur

ex: P (at least 1)=1-P(None)

General addition rule

P(A or B)= P(AUB)= P(A)+P(B)-P(A and B)

Mutually exclusive (Disjoint events)

Events that do not overlap

The events have no outcomes in common and therefore can never occur together

If A and B are mutually exclusive then: P(A or B)= P(A)+P(B)

Sample w/ replacement

Card is returned to deck→ shuffled→ prob is the same

Sampling w/o replacement

Card is removed, prob changes

Union

The set of all outcomes that are A or B or both (Or) (P AUB)

Complement

A^C= not A→mutually exclusive/disjoint

Intersection

The set of all outcomes that are both A and B at the same time (and)

General addition rule

For any 2 events A and B

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

Valid Prob. Models

-Prob of event is # between 0 and 1

-All possible outcomes together must have a probabilities that add up to one

-Probability that an event does not occur is 1-minus the probability that the event does occur

Question Wording

And=overlapping box

Given=use the total of whatever comes after your line as your denominator

Or=add the relevant categories (subtract the over lap if needed

P(hit)→total hits/ grand total

P(Post)→total post-season/grand total

P(hit and post)→ overlapping box/grand total

P(hit given post)→overlapping box/post season row total

conditional probability

The probability of event A under the condition that we know event B has occured

Prob of A given B→ P(A|B) → P (A and B)/P(B)

Prob of B given A → P(B|A)=P (A and B)/P(A)

Conditional probability and independent

When knowledge that one event has happened does not change the likelihood that another event will happen

P(A|B)=P(A) and P(B|A)=P(B)

General Multiplication rule

For any 2 events A and B (can be disjoint or not, dont overlap)

P(A and B)=P(A)xP(B|A)

For both of the two events to occur, first one must occur and then given that the first one has occurred, the second must occur

Tree diagram

All possibilities of what your doing

The probability of reaching the end of any complete branch is the product if the probabilities written in its segments

Multiply to the end