Statistics Module 3

In a ser of known events, the probability of a specific event occurring ranges from:

0.0 to 1.0

experiment

a process or activity that generates observable and well-defined outcomes

sample point

each outcome of an experiment

sample space

the set of all possible outcomes an experiment can produce

same space written as:

S = {list of sample points}

event

a collection of sample points

probability

equal to the sum of the probabilities of the sample points in an event

probability distribution

lists the outcomes of a probability experiment along with the probability of each outcome

complement of an event

• the complement of an event A is defined to be the event consisting of all sample points that are not in A

  • denoted by A^c

P(A) =

1 - P(A^c)

P(A) + P(A^c) =

1

probability rules

Rule #1: the probability for each assigned outcome must be between 0 and 1, inclusive.

• 0 ≤ P(E_i) ≤ 1 for each i

Rule #2: The sum of the probabilities for all the outcomes must equal 1.

• P(E₁) + P(E₂) + P(E₃) + … + P(E_n) = 1

union of two events A and B:

• the event containing all sample points belonging to A or B or both

• denoted by A U B

addition law provides:

a way to compute the probability of event A, or B, or both A and B occurringa

addition law written as:

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

mutually exclusive events

if the events have no sample points in common

addition law for mutually exclusive events:

P(A U B) = P(A) + P(B) [P (A U B) = 0]

conditional probability

the probability of an event given that another event has occurred

conditional probability of A given B denoted as

P(A/B)

two methods for calculating or assigning probabilities

• classical method

• relative frequency method (empirical method)

classical method for assigning probabilities utilizes

counting techniques

classical method is used…

whenever all of the outcomes in the sample space are equally likely to occur

relative frequency method (empirical method) is used….

whenever actual data is available

formula for relative frequency method

P(E) = (number of time event E is observed/occurs) / (number of trials in the experiment)

• P(E) = N(E) / N(S)

  • N(E) is the number of outcomes in event (E)

  • N(S) is the number of outcomes in the sample space

multiplication rule used for:

multiple-step experiments

multiplication rule

If an experiment has a sequence of k steps with each step having a corresponding n_k number of possible outcomes, then the total number of possible outcomes is given by: Total outcomes = (n₁)(n₂)(n₃)…(n_k)

factorials

n= n(n-1)(n-2)…(2)(1)

r= r)r-1)(r-2)…(2)(1)

special definition for factorial 0!

0!=1

counting rule for combinations

the number of combinations is the number of ways of choosing r objects out of a set of n objects

• C(n,r)= n!/ r!(n-r)!

counting rule for permutations

P(n,r)= n!/(n-r)!

number of permutations

the number of ways of choosing and arranging r objects from n objects

event

• any subset of the sample space

• the collection of sample points that make up the event

complement

• the collection of points in the sample space that do not make up the event

• the points left over from taking the sample points of the event away

• compliment of event E is represented by E^c

P(A)

the probability of Event A

P(A^c)

the probability of the complement of Event A

The compliment rule: since we know that the probability that the sample place will occur is 1, we can also say:

P(A) + P(A^c) = 1

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

The union of two events, A and B, contains all the sample points that belong to A or B or both

A∪B

an intersection of two events, A and B, contains all the sample points that belong both A and B

(A∩B)

addition law of probability used to:

calculate the probability of the union of two events A∪B

equation for addition law of probability

P(A∪B) = P(A) + P(B) - P(A∩B)

the subtraction of the P(A∩B) from

P(A∪B) = P(A) + P(B) - P(A∩B)

removes…

…one set of the double counted probabilities

mutually exclusive events

• situation in which there are two events in a sample space that have no common sample points

• P(A∩B)=0

conditional probabilities

• the probability that event A will occur given that event B has occurred

• P (A|B)

the occurrence of one event will affect the probability of another event occurring

the probability of one event is dependent on the occurrence of another event

the probability of event A is dependent on the occurrence of event B (as equation)

P(A│B) ≠ P(A)

• if the event A is dependent on event B, then the probability of A given B has occurred is going to be different than the probability that event A will occur (without B occurring)

the probability of A occurring is independent of B occurring

P(A│B) = P(A)

two events A and B are independent if and only if:

(test for independence)

P(A∩B) = P(A) * P(B)