ap stats unit 4

Sample Spaces

For a random process, the sample space is the collection of all possible non-overlapping outcomes. The letter S is often used to represent the sample space.

Event

Collection of outcomes for the random process. Events are usually denoted with a single capital letter, like A, B, etc.

Probability of an event formula: P(A)=

(total number of outcomes in event A) / (total number of outcomes in sample space)

*A probability will always be a number between 0 and 1, inclusive. A PROBABILITY OF 0 MEANS THE EVENT IS IMPOSSIBLE!

*A PROBABILITY OF 1 MEANS IT IS A CERTAINTY (IT WILL ALWAYS OCCUR)

Interpreting Probability

Probabilities of events in repeatable situations can be interpreted as the relative frequency with which the event will occur in the long run.

For there to be a valid probability distribution, the sum of the probabilities have to equal to ...

ONE

Complements

The complement of an event A, is the event that A does not happen.

The complement of A is denoted by:

A prime or A^c

The probability of the complement of A is equal to 1 minus the probability of A:

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

MUTUALLY EXCLUSIVE AND DISJOINT MEAN THE SAME THING!

Mutually Exclusive Events...

CANNOT OCCUR/HAPPEN AT THE SAME TIME!

Venn Diagrams

Venn Diagrams can be used to represent probabilities in a visual form. Venn Diagrams often use circles/ovals to represent events. The portion where the graphs overlap is the INTERSECTION of the two events, and is represented by the notation (A - upside down u- B)

(A - upside down u- B) is the same thing as:

A AND B!

Mutually Exclusive Events

Mutually Exclusive events are events that cannot occur at the same time. There would be no intersection of the two events.

Joint Probability

The probability of the intersection of two events

Two events are mutually exclusive (disjoint), if they cannot occur at the same time. If two events are mutually exclusive, then the probability of their intersection is 0.

Conditional Probabilities: P( B|A)

"B given A" What is the probability that event B will occur given that event A has occured?

FOR CONDITIONAL PROBABILITY QUESTIONS, IT MAY BE EASIEST TO DRAW OUT A TREE DIAGRAM!

Multiplication Rule:

To determine the probability of "AND" (both events occuring) , we multiply the probability of one event by the probability of another.

P (B | A ) Formula:

P (A and B) divided by P(A)

THE ORDER OF THE CONDITIONAL STATEMENT MATTERS!

Independent Events

Events A and B are independent if, and only if, knowing whether or not event A has occurred (or will occur) does not change the probability that event B will occur.

Independent Events (Continued):

P (A | B) = P(A); P(B|A) = P(B)

General Multiplication Rule

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

ONLY FOR INDEPENDENT EVENTS

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

IF THERE IS REPLACEMENT: INDEPENDENT EVENTS

How to tell if two events are independent or not...

Two events are independent if:

P ( E | F) = P(E)

Two events are independent if:

P (E and F) = P(E) P(F)

Union of events

The probability that event A OR event B (or both) will occur is the probability of the union of A and B, denoted P( A - u - B)

Addition Rule

States that the probability that event A or event B or both will occur is equal to the probability that event A will occur, plus the probabiliity that event B will occur, MINUS THE PROBABILIITY THAT BOTH EVENTS A AND B WILL OCCUR (The overlay).

Addition Rule Denoted:

P (A - u - B) = P(A) + P(B) - P ( A - upside down u - B)

If all outcomes in a sample space are equally likely, then the probability of an event E occuring can be defined as:

(Number of outcomes in event E) / (Total number of outcomes in sample space)

Recap:

The probability of an event is a number 0-1 inclusive.

The probability of the complement of an event E, denoted by E prime, E^c or Not E is equal to:

1- P(E)

Recap:

Probabilities can be interpreted as the long-run relative frequency the event will occur is the random process is repeated many times.

Recap 2:

Two events are mutually exclusive (disjoint) if they cannot occur at the same time. If two events A and B are mutually exclusive, then P (A - upside down u- B) = 0.

Conditional Probability

Probability that an event will occur, given that another event has already occurred.

P (A | B)

P (A ∩ B) / P(B)

The probability of the intersection of two events can be found using the formula:

P (A ∩ B) = P(A) * P (B | A)

If two events are independent then:

- P (A | B) = P(A)

-P(B|A) = P(B)

-P (A ∩ B) = P(A) * P(B)

The probability of the union of two events can be found by the formula:

P ( A U B) = P(A) + P(B) -P (A ∩ B)

If the conditional probability P(A | B) is equal to the unconditional probability P (A), then...

The events A and B are independent.

The probability for the union of two events can be found by using:

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

Verifying that it is a valid probability distribution:

1. Does it include every possible (non-zero) outcome for shooting 3 free throws?

2. Are all probabilities between 0 and 1?

3. Do the probabilities add up to 1?

If events are independent:

P(A) = P(A | B) = P (A | B^ C)

If you draw WITHOUT replacement, the situation is not independent.

How to tell if there is a valid probability model?

All the probabilities add to one and are between 0 and 1

Random Variables

Numerical outcomes of random behavior.

For Example:

X= The number of children in a randomly selected household

W = The time (min) it takes a randomly selected person to run a mile.

Y = the number of dogs with a location chip out of 10 randomly selected dogs.

L = The length (cm) of a randomly selected person's index finger.

Discrete versus Continuous Random Variables: Discrete Variable

*A discrete random variable can only take a countable number of values. (There will never be a 1.25) THERE WILL ALWAYS BE SPACE ON THE NUMBER LINE.

X= The number of children in a randomly selected household.

Discrete versus Continuous Random Variables: Continuous Variable

A continuous random variable can take on an infinite number of values in an interval on a number line. NO SPACE ON THE NUMBER LINE!

W= time (in minutes) it takes a randomly selected person to run a mile.

Probability Distribution

A display of the entire set of values with their associated probability is called a: Probability Distribution.

Properties of a Probability Distribution:

1. Each value is associated with a probability from 0 to 1

2. The sum of all probabilities must be 1.

Description of a probability distribution should include:

- Shape

- Center

- Spread

Finding the mean value from a probability distribution formula:

Expected Value = Each individual X value * Probability of Each individual x value

*ADD ALL OF THEM TOGETHER TO GET THE MEAN

IF YOU GET A DECIMAL VALUE FOR YOUR MEAN, THIS IS HOW YOU SHOULD INTERPRET THAT VALUE:

In the long run, if many prairie dog litters are randomly selected, the average number of pups per litter will be about 2.66 pups.

Standard Deviation Equation:

Root ((Σ (X -mean)^2 times P(xi))- ADD THEM ALL TOGETHER TO GET THE STANDARD DEVIATION!!

What does the standard deviation mean?

Average number that each data value varies from the mean

STANDARD DEVIATION IS THE SQUARE ROOT OF THE VARIANCE!

Ux REPRESENTS THE

MEAN

OX REPRESENTS THE

STANDARD DEVIATION

WHEN YOU TRANSFORM VALUES, YOU MULTIPLY AND ADD THE MEAN VALUE, BUT YOU JUST MULTIPLY THE STANDARD DEVIATION VALUE. BASICALLY, MULTIPLICATION AND ADDITION CHANGES THE MEAN VALUE, BUT ONLY MULTIPLICATION CHANGES THE SD VALUES.

Formula for finding the mean of a sum:

Add the two means together! Easy you got it.

Formula for finding the difference of a sum:

Subtract the two means! Easy you got it.

Two random variables are independent if...

Knowing information about one of them does not change the probability distribution of the other.

If two random variables X and Y are independent then:

For a sum or difference of independent random variables, VARIANCES ADD. NOTE THAT STANDARD DEVIATIONS, HOWEVER, NEVER ADD!

VARIANCE IS THE SAME THING AS:

THE STANDARD DEVIATION SQUARED.

TO FIND THE STANDARD DEVIATION:

1) Add the two variances together.

2) Square root the sum of the two variances

- This gets you the standard deviation!!!

For any two random variables X and Y, and real numbers a and b, the expression aX + bY is called a linear combination of X and Y

- Mean: a(ux) + b(uy)

-Standard Deviation: