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