CSET: Math II - Probability

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Laws of Probability

* A probability of zero means that the event cannot happen
* A probability of one means that the event must happen.
* All probabilities must be between zero and one inclusively.
* The sum of probabilities of all simple events must be equal to one

Compound Event

An event defined by combining two or more events.

The union of events A and B

is the set of outcomes that are included in A or B or both, and is denoted as A ⋃B.

The intersection of the events A and B

is the set of all outcomes that are included in both A and B.
Notation: A ∩ B

The complement of an event A

is denoted AC and refers to the set of all outcomes in the sample space that are not in A.

Two sets are mutually exclusive if

they have no points in common.

The probability of AC

is given by P (A^C) = 1 - P (A)

If the events A & B are mutually exclusive, then:

P (A ⋃B) = P(A) + P(B)

For any two events A and B,
the P(A ⋃B) =

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

Conditional Probability

* The probability that one event will occur, given that some events have occurred or is certain to occur.
* The conditional probability of A, given that B has occurred is (P(A|B))=P(A∩B)/P(B)

Independence

Two events, A and B, are independent if and only if the P(A | B)= P(A) or P(B | A) = P(B)
Example: Tossing a coin and picking a card are independent events.

Multiplication Rule for Independent Events

If two events, A and B are independent, then P (A and B) = P(A)* P(B)

Random Variable:

A variable that can take on different values according to the outcome of an experiment.

Discrete Random Variable:

A random variable that can take on only certain values along an interval, with the possible values having gaps between them. Example: In a given group of 5 children, the number who got at least one electronic toy for Christmas would be 0, 1, 2, 3, 4, 5. It could not be a number between any of these values.

Continuous Random Variable:

A random variable that can take on a value at any point along an interval. Example: The exact temperature outside your classroom could be 75.5°F, 80.2° F, or anything in-between.

Discrete Probability Distribution

A discrete probability distribution is a listing of all possible outcomes of an experiment, along with their respective probabilities of occurrence.

Example: The probability of tossing 4 coins and getting a head.
To answer this, set up the set of all possible results of tossing four coins: (H,H,H,H), (H,H,H,T), (H,H,T,H), (H,H,T,T), (H,T,H,H), (H,T,H,T), (H,T,T,H), (H,T,T,T), (T,H,H,H), (T,H,H,T), (T,H,T,H), (T,H,T,T), (T,T,H,H), (T,T,H,T), (T,T,T,H), (T,T,T,T). From this, we can see that the possibility is 1:16.

Characteristics of a Discrete Probability Distribution

. For any value of x, 0<1.
2. The values of x are exhaustive: The probability distribution includes all values.
3. The values of x are mutually exclusive: only one value can occur for a given experiment.
4. The sum of their probabilities is one, or P(x¯)=1.0

The Classical Approach

Describes probability in terms of the proportion of times that an event can be theoretically expected to occur. Here the probability of success is based upon the prior knowledge of the process involved. Probability =XT, where X= the number of possible outcomes in which the event occurs and T= the total number of possible outcomes

The Relative Frequency Approach:

Is the proportion of times an event is observed to occur in a very large number of trials. This approach is also known as, Empirical Classical Probability

Probability =XTwhere X= the number of possible outcomes in which the event occurs and T= the total number of possible outcomes.

When events are mutually exclusive:

The occurrence of one means that none of the others can occur.
In this case, the probability that one event will occur is the sum of their individual probabilities.
P(AorB)=P(A∪B)=P(A)+P(B).

The probability of picking a Queen or a King from a standard deck of 52 cards = 1/13+1/13=2/13.

When events are not mutually exclusive:

Two or more of them can happen at the same time.
P(AorB)=P(A∪B)=P(A)+P(B)-P(A∩B). Recall that P(A∩B)=P(AandB)

Example: The probability of picking a Jack or a diamond from a standard deck of 52 cards = 4/52+13/52-1/52=16/52
P(A and B) = P(A) X P(B|A)

Binomial Distribution Characteristics

Binomial distribution deals with consecutive trials, each of which has two outcomes, H/T; Yes/No, etc..

Characteristics

* There are two or more consecutive trials.
* In each trial, there are just two possible outcomes - usually denoted as success or failure
* The trials are statistically independent. That is, the outcome of one trial is not affected by the outcomes of the earlier trials, and it does not affect the outcomes of later trials
* The probability of success remains the same from one trial to the next.
If n = number of trials
And k = number of successes
And p = probability of success:

Poisson Distribution

Poisson distribution is a discrete probability distribution that is applied to events for which the probability of occurrence over a given span of time, space, or distance is extremely small.

Poisson distribution describes:

* Customer arrival at a service point during a given period of time - such as the number of motorists approaching a toll booth, or the number of hungry persons entering a McDonald's restaurant, or the number of calls received by a company's switchboard
* Defects in manufactured materials - such as the number of flaws in wire or pipe products over a given number of feet, or the number of knots in wooden panels for a given area
* The number of work-related deaths, accidents, or injuries over a given number of production hours
* The number of births, deaths, marriages, divorces suicides, and homicides over a given period of time

Fundamental Counting Principle

In a sequence of events, the total possible number of ways all events can be performed is the product of the possible number of ways each individual event can be performed.

Example: How many 7-symbol license plates are possible if the first symbol must be the letters A , B, or C, the second symbol any letter of the alphabet, and the other 5 are digits?

3 x 26 x 10 x 10 x 10 x 10 x 10 = 7,800,000

Factorials

If n is a positive integer, then

n! = n (n-1) (n-2) ... (3)(2)(1)
n! = n (n-1)! A special case is 0! note 0! = 1

Permutations

A permutation is an arrangement of objects without repetition where order is important. You can think of the example of a parade, where we would consider it a different arrangement of the parade if the same people participated, yet marched in a different order, no matter how slight the difference in order.

Permutations using all the objects
A permutation of n objects:
nPn = P(n,n) = n!

Example: Find all permutations of the letters "ABC"
Example: Alfred, Bertha, Chandra, Doug, and Eve are forming a parade. How many ways can they march?
{ABC, ACB, BAC, BCA, CAB, CBA}
A permutation of n objects, arranged in groups of size r, without repetition, and with order being important is:
nPr = P(n,r)=n!/(n-r)!

Arrangements when objects are not distinguishable:

To arrange the letters in the word MISSISSIPPI:

We could think of this first as arranging 11 distinguishable objects: 11!
Then divide out all the duplications: 4 I's, 4 S's, 2 P's

11!/4!4!2!

Combinations

A combination is an arrangement of objects without repetition where order is not important. You might think of the example of committees, where what is of interest is only who the collection of individuals are.
Note: The only difference in the definition of a permutation and a combination is whether order is important.

A combination of n objects, arranged in groups of size r, without repetition, and order being important is:
nCr = C(n,r) = n! / ( (n-r)! * r! )

Example: Find all two-people teams (committees) that can be formed between: Alvin, Bertha and Charles.
AB = BA AC = CA BC = CB

There are only three different teams that can be formed.

Example: Five people want to be on a committee, but only three can. How many possible committees can be formed?
5C3 = 10

Order Doesn't Matter, Balls not Replaced

n!/r!(n-r)!

Order Does Matter, Balls not Replaced

n!/(n-r)!

Order Doesn't Matter, Balls Are Replaced

(n+r-1)!/r!(n-1)!

Order Does Matter, Balls are ReplacedP

n^r

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