AP Statistics: Module 4, combined

00disjoint

(of two or more sets) having no elements in common Ex: P(A U B) = P(A or B) = P(A) + P(B)

When conducting a simulation, there are five steps to remember:

-State the problem or question. -State the assumptions. -Describe the process for one repetition, including possible outcomes, assigned representations, and measured variables. -Simulate many repetitions. -State conclusions.

P(A|B)

Probability of event A given that event B occurs.

expected value

the sum of the products of each possible value and the probability that it occurs; takes into account that the various outcomes may not be equally likely E(x)=sum of x*P(x), the average/mean outcome after repeated events. E(X) = μx = Σ [ xi * P(xi) ]

independent

the trial does not depend on past/future trials; knowing that one occurs does not change the probability that the other occurs Ex: the chance of landing on heads is always 50%. Each flip has no bearing or influence on the next flip

term for using the law of large numbers to predict what will happen in the long run

theoretical probability

00continuous random variable

there must be an infinite number of outcomes

00conditional probability

used to find the probability of an event given another event has occurred; it simply means one event depends on another; used to prove whether two events are independent—meaning, the outcome of one event does not influence the probability of the other.

independent event

when the occurrence of one event does not change the probability that the other event will happen

geometric probability density function

when you are calculating the exact probability for a specific value

probability of heads

H = 0.5 or 1/2

00General Addition Formula

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

00General Multiplication Rule

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

Sample Space

The collection of all possible outcome values

00Binomial random variable

The number of successes, x, in repeated trials of a binomial experiment.

00event

any outcome or collection of outcomes that is a subset of the sample space

A binomial experiment must meet certain requirements...

-Each outcome is either a success (P) or a failure (Q). -All trials are independent. -There are a fixed number of "n" trials. -The probability of success, p, is the same for each trial.

Rules to Remember

-For any event A, the probability (P) must be between zero (event can't occur) and one (event will always occur): 0 ≤ P(A) ≤ 1. -The sum of all possible outcomes (sample space [S]) of a trial must be equal to one: P(S) = 1. -Complement Rule: The probability of an event occurring is one less the probability that it does not occur. This is known as the complement of A (or Ac or A'): P(A) = 1 - P(Ac).

Examples of Binomial Distributions

-The number of male or female births in the next 20 births at a local hospital -The number of successful free throws in the next 30 free throws -The number of correct answers on a 10-question multiple-choice exam when guessing at the answers -The number of people who show up for a flight when 100 tickets are sold

Examples of sample spaces

-Tossing a coin: S = {H, T}, where H is heads and T is tails Tossing a coin twice: S = {HH, TT, HT, TH} -Rolling a die: S = {1, 2, 3, 4, 5, 6} Rolling two dice: S = {1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 2 1, 2 2, 2 3, 2 4, 2 5, 2 6, 3 1, 3 2, 3 3, 3 4, 3 5, 3 6, 4 1, 4 2, 4 3, 4 4, 4 5, 4 6, 5 1, 5 2, 5 3, 5 4, 5 5, 5 6, 6 1, 6 2, 6 3, 6 4, 6 5, 6 6}

Addition Rule/ P(A or B)

-Two events are disjoint, or mutually exclusive, if they have no outcomes in common. For example, the probability of being both a junior and a senior in high school is zero because it is not possible to be both at once. P(A) + P(B) Must be disjoint (mutually exclusive)

00If done correctly, all possible outcomes must add up to _______.

1 (100%)

Mean (Ex) of a geometric random variable

1 / p

Probability is always between what two numbers?

1 and 0

4 steps of running a simulation

1) Clearly indicate what you are trying to find 2) Plan the way you will use numbers 3) Carry out trials 4) Draw conclusions

2 main parts of a probability model

1) Sample space 2) probability for each outcome

Important parts of planning how you will use your numbers in a simulation

1) what numbers will be used to reflect the situation 2) what does a trial look like 3) what makes a trial successful

00complement of A

1-P(A)

P(at least 1)

1-P(none)

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)

The "And" Rules

1. For independent events: P(A and B)=P(A)xP(B) 2. General Rule for Indep./Non-Indep.: P(A and B)=P(A)xP(B/A)

The "Or" Rules

1. If the events are disjoint: P(A or B) = P(A) + P(B) 2. If the events are not disjoint: P(A or B) = P(A) + P(B) - P(A and B)

Steps in simulation: (5 Steps)

1. State problem, 2. State assumptions, 3. Describe the process for 1 repetition, 4. Simulate many repetitions, 5. State conclusions.

What is the number of outcomes in the sample space of tossing a coin eight times?

28 = 256

What is the number of outcomes in the sample space of rolling a die four times?

64 = 1,296

What is the number of outcomes in the sample space for choosing from eight sandwich options over five visits to the sandwich shop?

85 = 32,768

With Replacement Without Replacement

A deck of cards: means the cards put back in the original set and the probabilities of a single draw remain the same A deck of cards: the 1st card since it is not replaced the probabilities are different for the 2nd choice

Probability Model

A description of some chance process that consists of two parts: a sample space S and a probability for each outcome.

Tree Diagram

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

Law of Averages

A flawed way of thinking that if an event doesn't happen for a while that it is due to occur soon.

Probability

A number between 0 and 1 that describes the proportion of times any outcome of a chance process would occur in a very long series of repetitions.

Simulation

A random process of numerous trials used to estimate probability and imitate chance behavior

Trial

A single attempt

Two Way Table

A table that lists counts for 2 categorical variables that is helpful with conditional probabilities.

Venn Diagram

A way to illustrate the sample space of a chance process including two events, consisting of two circles representing the events.

00Combining random variables

Add means, first convert standard deviations to variances , add and take square root

Transforming variables by addition/subtraction

Add/subtract that value from the mean, leave standard deviation alone

probability distribution - properties

All probabilities are between 0 and 1 inclusive sum of all probabilities equals 1

Intersection (∩)

All the outcomes in common between two events compared. P(A and B)

Union (U)

All the outcomes in the two events included. P(A or B)

Universal Set

All the students included in the data make up the universal set. The universal set is the set that contains all elements of the sample space. In this instance, the universal set is 990, representing all students in the school.

Random Experiment

An activity whose outcome we can observe or measure but cannot predetermine

Independent event

An event whose result does not depend on the result of another event

Simulation

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

Simulation

An replication of repeating a procedure a large number of times

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)

00Complement

Another subset of the original set is called the complement. The complement of a set contains all elements in the universal set that are not in that particular set. In the diagram above, the complement of set A is all students in set E who are not taking music, which would be 383 students (990 − 265 − 342 = 383). The complement of set A is written as A'.

00Event

Any collection of outcomes from some chance process. A subset of the sample space. Usually designated by capital letters.

Law of Large Numbers

As the number of experiments increases, the actual ratio of outcomes will converge on the theoretical, or expected, ratio of outcomes

Law of Large Numbers

As the number of trials increases, the probability approaches its theoretical value (e.g. role a dice 4 times vs. rolling it 400 times)

00Binomial Random Variable Notation

B(n, p)

Cumulative Binomial Distribution

Calculates that probability that x occurs less than equal to k number of times

00Discrete Random Variables

Can list all possible outcomes (in a probability model)

00Continous Random Variables

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

Tree Diagram

Displays the sample space of a process involving a sequence of events, with each each subsequent event branching out from the first like a tree.

Combining Random Variables T = (X + Y) or (X - Y)

E(T): E(X) + E(Y) or E(X)-E(Y) S(T): Square root [Var(X) + Var(Y)] Variances always add

00Expected value of x

E(X) = μx = Σ [ xi * P(xi) ]

00Expected Value

E(x)=sum of x*P(x), the average/mean outcome after repeated events.

Independent event

Event that does not affect another; each trial is independent from the others (rolling a dice)

"At least one"

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

General Multiplication Rule

Finds the probability both A and B occur using the formula: P(A and B) = P(A∩B) = P(A) * P(B|A)

Probability Example

For instance, if we flip a coin once, the proportion of tosses that comes up heads is either one (100%) or zero (0%). However, as we continue to toss the coin more and more, the proportion of tosses that comes up heads will approach 0.5 (50%). This shows how the empirical probability (the probability in actual trials) approaches the theoretical probability (the probability that is calculated using a formula).

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.

Multiplication Rule for Independent Events

If A and B are independent, probability A and B both occur is: P(A∩B) = P(A) * P(B)

Independence

If knowing whether one event occurs does not alter the probability that the other event occurs.

Success/Failure Condition

If np>10 and nq>10, then we can use normal model for binomial probability.

Law of Large Numbers

If we observe more and more repetitions of any chance process, the proportion of times a specific outcome will occur approaches a single value (in the long run). In the short run, it is unpredictable.

Probability Distribution

Lists all possible outcomes of a random event and each outcome's probabilities

Law of Large Numbers

Long-run relative frequency of repeated independent events becomes close to actual frequency as number of trials increases.

∩

Multiplication

Transforming variables by multiplication/addition

Multiply/divide the mean by that variable, multiply/divide the variance by the square (^2) of that variable)

00Disjoint

Mutually exclusive events

00Continuous random variables

Not whole numbers, obtained by measuring

Tree Diagram

One type of diagram that is useful when solving conditional probability problems is a tree diagram. These diagrams organize possible outcomes into increasingly specific combinations of outcomes. The goal is to have an accurate list of all events in the sample space to use in calculating probabilities.

Conditional Probability, P(A|B)

Only used on dependent events; equals P(A n B)/P(B)

Independence

Outcome of one trial does not influence the outcome of another. P(A|B) = P(A)

probability of rolling a 3 or 2 on a single die

P(3) = 1/6, P(2) = 1/6: P(3 or 2) = 1/6 + 1/6 = 1/3 since they a mutually exclusive

None/At Least

P(A Complement)=1-P(A)

00General Addition Rule (mutually exclusive events)

P(A U B) = P(A) + P(B)

00General Addition Rule (not mutually exclusive)

P(A U B) = P(A) + P(B) - P(A intersect B)

multiplication rule

P(A and B) = P(A∩B) = P(A)P(B)

00General Addition Rule

P(A or B) = P(A) + P(B) - P(A and B) Fixes the double counting problem because of the overlapping outcomes.

General addition rule

P(A or B) = P(A) + P(B) - P(A and B) OR P(A U B)= P(A) + P(B) - P(A and B) (removes duplication) If A and B are not mutually exclusive: P(A or B) = P(A) + P(B) - P(A and B) If A and B are mutually exclusive: P(A or B) = P(A) + P(B) P(A and B) = 0

Multiplication Rule of Independent Events

P(A) * P(B)

P (A U B)

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

P(A) (where P(A) is read as "the probability of A.")

P(A) = Number of times A occurs / Total number of outcomes

P(A and B) for independent events

P(A) x P(B)

P(A and B) for dependent events

P(A) x P(B|A)

00disjoint addition rule

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

00general addition rule

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

reverse conditioning (baye's rule)

P(A|B) = P(B|A)*P(A)/P(B)

Multiplication rule of dependent events

P(A|B)*P(B) = P(A intersect B)

00Conditional Probability

P(A|B)=P(A and B)/P(B). P(B | A) = P(A intersect B)/P(A)

Independent Multiplication Rule

P(A∩B)=P(A)*P(B)

Not Independent Multiplication rule

P(A∩B)=P(A)*P(B|A)

Mutually Exclusive Addition Rule

P(A∪B)=P(A)+P(B)

Non Mutually Exclusive Addition Rule

P(A∪B)=P(A)+P(B)-P(A∩B)

00Conditional Probability

P(B | A) = P(A intersect B)/P(A)

00Complement rule

Probability that an event does not occur equals 1 - probability that it does

00Complement

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

Binom Cdf (10, 0.5, 5)

Probability that x occurs less than OR equal to 5 times, in 10 trials

1 - Binom Cdf (10, 0.5, 5)

Probability that x occurs more than 5 number of times in 10 trials

Geometric cdf

Probability that you have to run k or fewer trials to get 1 success

Geometric pdf

Probability that you have to run k trials to get 1 success

Sample Space (Theoretical population)

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

Venn Diagram

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

Tree Diagram

Shows all possible outcomes and probability for all possible outcomes

geometric random variable

Special type of discrete random variable that counts the number of trails until the first success 1. Independent trials, 2. Same probability, 3. Success or failure, 4. Run trials until one success is achieved.

Binomial Coefficient on Calc

Step 1: Type the value of n on the main screen of your calculator Step 2: Select [MATH], then choose PRB and scroll down to 3:nCr Step 3: Press [ENTER], then enter the value of k or r. Last, press [ENTER] again to calculate

2nd Dist Binompdf(n, p, x) 2nd Dist Binomcdf(n, p, x)

Steps to get cumulative binomial probability on TI Steps to get cumulative binomial probability on TI

Something has to happen rule

Sum of all probabilities of all possible outcomes is 1

Probability formula that works even if the events are not mutually exclusive

The Addition Rule P (A or B) = P(A) + P(B) - P(A and B) *when mutually exclusive, P(A and B) = 0

Intersection

The area in the center that overlaps is called the intersection. The intersection of two sets is the set that contains elements that are common to both sets. Intersections are written mathematically as A ∩ B. The intersection of sets A and B is the students who are taking both music and drama—in other words, the number of students in the overlapping portion of the Venn diagram. There are 342 students in A ∩ B.

Union

The area that is included in the two circles is called the union. The union of two sets is the set that contains all elements of those two sets. In other words, all numbers within both circles in the Venn diagram. The union is written mathematically as A U B. In the diagram above, A U B includes 900 students—the 265 music students, the 293 drama students, and the 342 students taking both music and drama. Notice that this set does not include the 90 students who do not want to take music or drama. If those students were included, you would be looking at the universal set.

00Complement

The complement of any event A is the probability that event A will not occur. If the probability that it will snow today is 40%, then the probability that it will not snow is 60%.

Binomial Random Variables

The number of successes, x, in repeated trials of a binomial experiment. -Number of times x occurs; 1. Independent trials, 2. Each trials is a success or failure, 3. Set number of trials, 4. Probability is equal for all trials.

Dependent event

The outcome of 1 event affects the outcome of the 2nd event (e.x. choosing cards w/o replacement)

Probability Rules

The probability P(A) of any event is always a number between zero and one. If S is the sample space in a probability model, then P(S) = 1. That is, the probability that you will get something from the sample space is one—meaning, it will definitely happen. -Complement -Formula

Conditional Probability

The probability an event will occur given another event has already occurred. it simply means one event depends on another; used to prove whether two events are independent—meaning, the outcome of one event does not influence the probability of the other. Denoted by P(B|A). P(B|A)=P(A∩B)/P(B) or P(A andB)= P(A|B)P(B) or P(B | A) = P(A intersect B)/P(A) P(A|B)= P(B∩A)/P(A) or P(A and B)P(A|B)P(A)

00 Binomial distribution

The probability distribution of a binomial random variable.

Probability Example

The probability of Joe getting to school before the tardy bell rings is about 80%. What is the probability that Joe does not get to school on time? P(Not on time) = 1 - P(On time) = 1 - 0.8 = 0.2 There is a 20% chance that Joe will not get to school on time.

00conditional probability

The probability of an event can change if we know some other event has occurred. Ex: what is the probability of getting a heart from a standard deck of cards?

00 Binomial Probability

The probability of getting one of two outcomes (like heads or tails)

Binomial Probability

The probability of getting one of two outcomes (like heads or tails) Counts number of success (X or k equals number of successes) with fixed number of Bernoulli Trials (n=number of trials).

Theoretical probability

The probability that is calculated using a formula.

00Complement

The scenario where an event does NOT occur. The _______ of event A, is calculated 1 - P(A)

Sample Space (S)

The set of all possible outcomes of a chance process.

Sample Space

The total possible outcomes

00Disjoint (Mutually Exclusive)

They share no outcomes in common.

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.

Disjoint/Mutually exclusive

Two events do not share outcomes. Therefore, if A occurs, B cannot occur. P(A and B) = 0

Independent Events

Two events in which the occurrence of one event does not change the probability that the other with happen. P(A|B) = P(A), and P(B|A) = P(B)

Mutually Exclusive (Disjoint)

Two outcomes that have no outcomes in common so can never occur together. These events can never be independent, because if one can't happen with the other. If one happens, the other can never happen.

Random Variable

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

Outcome

Value measured, observed, or reported for an individual instance of the trial

Independent event

When the occurrence of one event does not change the probability that the other event will happen

00Conditional probability

When we want to find the probability of an event given another event has occurred, we refer to this as conditional probability. A conditional probability is written as P(A|B), which reads "the probability of A given B". We can use a formula to determine conditional probability.

00Discrete random variables

Whole numbers obtained by counting

00Geometric Random Variables

X occurs 1 time; 1. Independent trials, 2. Same probability, 3. Success or failure, 4. Run trials until one success is achieved.

venn diagram

a helpful diagram to draw when you have events that are not mutually exclusive.

sample space

a listing of all possible outcomes for a random situation

simulation

a random process of numerous trials used to estimate probability and imitate chance behavior

A method of determining mathematically the number of outcomes in a sample space when...

a trial is repeated is the number of outcomes of one trial raised to the power of the total number of trials.

random variable

a variable whose value is a numerical outcome from a random event; to be discrete, there must be a countable number of outcomes Ex: a variable whose value is a numerical outcome from a random even

00disjoint

aka mutually exclusive

binomial probability density function

allows us to find the probability of any individual outcome - Ex: Suppose a high school tennis player has a 12% probability of getting a serve inside the lines. What is the probability that out of 20 tennis serves that she makes, exactly four will get in?

binomial cumulative distribution function

allows us to find the probability of combined outcomes up to the x value.

union

an event where at least one of the two possibilities will occur; may occur in various combinations, but one of the possibilities must occur ; the overlap of two events noted by AUB

event

any outcome or collection of outcomes which are a subset of the sample space

trial

any procedure that can be repeated infinitely and that has a well-defined set of possible outcomes

law of large numbers

as we observe more and more repetitions of any chance process, the proportion of times a specific outcome occurs approaches a single value.

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)

any situation where the outcome is unknown

chance process

probability that something will not occur

complement of the probability of the event occurring

tree diagrams

diagram used in strategic decision making, valuation or probability calculations; starts at a single node, with branches emanating to additional nodes, which represent mutually exclusive decisions or events

disjoint / mutually exclusive

displayed when two events are mutually exclusive and they have no outcomes in common Ex: P(A U B) = P(A or B) = P(A) + P(B) OR P(AUB) = P(A or B) = P(A) + P(B)

A collection of outcomes

event; denoted with a capital letter

term for exploring the outcome of an event after a specific number of trials

experimental probability

multiplication principle

if there are a ways of doing something and b ways of doing another thing, then there are a. · b ways of performing both actions Ex: 4 sandwiches x 2 sides x 3 drinks gives you a total of 24 different choices.

when is a chance process predictable?

in the long run

when is a chance process unpredictable?

in the short run

The sample space (S)...

is a listing of all possible outcomes for a random situation.

An event...

is any outcome or collection of outcomes that is a subset of the sample space.

when two events cannot occur simultaneously we say they are ________________________

mutually exclusive

proportion of times an outcome will occur after a very long series of trials

probability

model that shows what outcomes are possible

probability model

P(B|A)

probability of B given that A occured

discrete random variables

random variables with a countable number of outcomes; their value is obtained by counting

continuous random variables

random variables with outcomes that can take on any numeric value within the range of values; their value is obtained by measuring

00complement

set contains all elements in the universal set that are not in that particular set.

complement

set contains all elements in the universal set that are not in that particular set. -the probability that the event A will not occur -is calculated 1 - P(A) Ex: S^c = it will not snow P(S) = 0.4 P(S^c) = 1 - P(S) = 1 - 0.4 = 0.6

pretending to perform a chance behavior using numbers to represent the scenario

simulation

Standard deviation

square root of [(x1 - mean1)^2 + (x2 - mean2)^2 + (x3 - mean3)^2]

Law of Large Numbers...

states that, as we observe more and more repetitions of any chance process, the proportion of times a specific outcome occurs approaches a single value.

00how do we find the complement of an event?

subtract the probability of the event occurring from 1

00complement of A

the event consisting of all sample points that are not in A; the probability of an event occurring is one less the probability that it does not occur.

Probability

the likelihood that an event will occur; the mathematics of chance -The probability of any outcome of a chance process is a value between zero and one. -A probability of zero means there is no chance of an event happening. -A probability of one means there is a 100% chance of an event happening.

00binomial random variable

the number of successes, x, in repeated trials of a binomial experiment

Benford's Law

the principle that in any large, randomly produced set of natural numbers, such as tables of logarithms or corporate sales statistics, around 30 percent will begin with the digit 1, 18 percent with 2, and so on, with the smallest percentage beginning with 9

00 binomial distribution

the probability distribution of a binomial random variable

00empirical probability

the probability in actual trials

Empirical probability

the probability in actual trials.

theoretical probability

the probability that is calculated using a formula

sample space (s)

the set of all possible outcomes; is a listing of all possible outcomes for a random situation

universal set

the set that contains all elements of the sample space

union

the set that contains all elements of those two sets. In other words, all numbers within both circles in the Venn diagram. The union is written mathematically as A U B.

intersection

the set that contains elements that are common to both sets. Intersections are written mathematically as A ∩ B.

Variance

the square of the standard deviation

standard deviation

the square root of the variance

Mutually exclusive

when two events do not have any outcomes in common and cannot happen at the same time i.e. drawing a 6 and 7 card P(A∪B)=P(A)+P(B)

00geometric cumulative distribution function

when you are calculating the accumulated probabilities for one up to a certain value given

sample space of a die

{1,2,3,4,5,6}

sample space of a coin toss

{Head, Tail}

combined mean

μx + y = μx + μy