Concepts of Probability Introduction to Probability & Statistics Concepts of Probability

Probability Concepts S = Sample Space : the set of all possible unique outcomes of a repeatable experiment. Ex: flip of a coin S = {H,T} No. dots on top face of a die S = {1, 2, 3, 4, 5, 6} Body Temperature of a live human S = [88,108]

Probability Concepts Event : a subset of outcomes from a sample space. Simple Event: one outcome; e.g. get a 3 on one throw of a die A = {3} Composite Event: get 3 or more on throw of a die A = {3, 4, 5, 6}

Rules of Events Union : event consisting of all outcomes present in one or more of events making up union. Ex: A = {1, 2}B = {2, 4, 6} A  B = {1, 2, 4, 6}

Rules of Events Intersection : event consisting of all outcomes present in each contributing event. Ex: A = {1, 2}B = {2, 4, 6} A  B = {2}

Rules of Events Complement : consists of the outcomes in the sample space which are not in stipulated event Ex: A = {1, 2}S = {1, 2, 3, 4, 5, 6} A = {3, 4, 5, 6}

Rules of Events Mutually Exclusive : two events are mutually exclusive if their intersection is null Ex: A = {1, 2, 3}B = {4, 5, 6} A  B = { } = 

Probability Defined u Equally Likely Events If m out of the n equally likely outcomes in an experiment pertain to event A, then p(A) = m/n

Probability Defined u Equally Likely Events If m out of the n equally likely outcomes in an experiment pertain to event A, then p(A) = m/n Ex: Die example has 6 equally likely outcomes: p(2) = 1/6 p(even) = 3/6

Probability Defined u Suppose we have a workforce which is comprised of 6 technical people and 4 in administrative support.

Probability Defined u Suppose we have a workforce which is comprised of 6 technical people and 4 in administrative support. P(technical)= 6/10 P(admin) = 4/10

Rules of Probability Let A = an event defined on the event space S 1.0 < P(A) < 1 2.P(S) = 1 3.P( ) = 0 4.P(A) + P( A ) = 1

Addition Rule P(A  B) = P(A) + P(B) - P(A  B) AB

Addition Rule P(A  B) = P(A) AB

Addition Rule P(A  B) = P(A) + P(B) AB

Addition Rule P(A  B) = P(A) + P(B) - P(A  B) AB

Example u Suppose we have technical and administrative support people some of whom are male and some of whom are female.

Example (cont) u If we select a worker at random, compute the following probabilities: P(technical) = 18/30

Example (cont) u If we select a worker at random, compute the following probabilities: P(female) = 14/30

Example (cont) u If we select a worker at random, compute the following probabilities: P(technical or female) = 22/30

Example (cont) u If we select a worker at random, compute the following probabilities: P(technical and female) = 10/30

u Alternatively we can find the probability of randomly selecting a technical person or a female by use of the addition rule. = 18/ / /30 = 22/30 Example (cont) )()()()(FTPFPTPFTP  -+= 

Operational Rules Mutually Exclusive Events: P(A  B) = P(A) + P(B) AB

Conditional Probability Suppose we look at the intersection of two events A and B. AB

Conditional Probability Now suppose we know that event A has occurred. What is the probability of B given A? A A  B P(B|A) = P( A  B)/P(A)

Example u Returning to our workers, suppose we know we have a technical person.

Example u Returning to our workers, suppose we know we have a technical person. Then, P(Female | Technical) = 10/18

Example u Alternatively, P(F | T) = P(F T) / P(T) = (10/30) / (18/30) = 10/18 

Independent Events u Two events are independent if P(A|B) = P(A) or P(B|A) = P(B) In words, the probability of A is in no way affected by the outcome of B or vice versa.

Example u Suppose we flip a fair coin. The possible outcomes are HT The probability of getting a head is then P(H) = 1/2

Example u If the first coin is a head, what is the probability of getting a head on the second toss? H,H H,T T,HT,T P(H 2 |H 1 ) = 1/2

Example u If the first coin is a head, what is the probability of getting a head on the second toss? H,H H,T T,HT,T P(H 2 |H 1 ) = 1/2 = P(H 2 ) Tosses are independent

Multiplication Rule P(B|A) = P( A  B)/P(A) P(A  B) = P(A)P(B|A)

Multiplication Rule P(B|A) = P( A  B)/P(A) P(A  B) = P(A)P(B|A) Independence : P(B|A) = P(B) P(A  B) = P(A)P(B)

Example u Suppose we flip a fair coin twice. The possible outcomes are: H,H H,T T,HT,T P(2 heads) = P(H,H) = 1/4

Example u Alternatively P(2 heads) = P(H 1  H 2 ) = P(H 1 )P(H 2 |H 1 ) = P(H 1 )P(H 2 ) = 1/2 x 1/2 = 1/4

Example u Suppose we have a workforce consisting of male technical people, female technical people, male administrative support, and female administrative support. Suppose the make up is as follows Tech Admin Male Female

Example Let M = male, F = female, T = technical, and A = administrative. Compute the following: P(M  T) = ? P(T|F) = ? P(M|T) = ? Tech Admin Male Female

Example Let M = male, F = female, T = technical, and A = administrative. Compute the following: P(M  T) = 8/30 P(T|F) = 10/14 P(M|T) = 8/18 Tech Admin Male Female