Examples

1.      Toss a fair coin twice

S = {HH, HT, TH, TT }

2.     Count the number of cases of equine West Nile virus from June 1,

2020 to May 31, 2030.

S = {0,1, 2, .....}

3.       Record the time to failure of an electrical component.

S = {t  0 < t <∞ }

4. Record the number of correct answers a student gets on the Math and Reading STARR exam.

S = {(x, y)  x = 0,1, ..., 40 ; y = 0,1, 2, ..., 40}

2.2 Probability Axioms

Probability Rules

■     Empty Set

■     P(AnB’)

■     Complement

■   General Additivity(Two Events)

■   General Additivity (3 Events)

■   What about more than 3 events? 

Example:  Suppose that 18% of U.S . residents have traveled to

Canada, 9% have traveled to Mexico and 4% have traveled to both countries. Find the probability that a randomly selected individual has

a) traveled to at least one of the two countries

b) traveled to Mexico, but not Canada

c) traveled to neither Canada or Mexico

Example  2:    Suppose  stocks  of  companies  A,  B,  and  C  are

popular.   Suppose  20% of  people own A stocks, 49% own  B stocks, 32% own C stocks, 8% own A and B stocks, 12% own A and C stocks, 10% own B and C stocks, and 5% own all 3 stocks.

Find  the  probability  that  a  randomly  selected  investor  owns ■   At least one of these stocks.

■   None of these stocks.

■   Only stock A.

■   Stock A and B, but not stock C.

How do we assign probabilities?

■  Subjective Approach

■  Relative Frequency Approach

■  Classical Approach with equally likely outcomes.

Calculating the probability of an event using

the relative frequency approach

■  A study on cocaine by gender and frequency in The American Journal of Drug and Alcohol Abuse gives the following data.

1.    Suppose we pick a person at random from this sample.

What is the probability that this person will be male?

2.   Suppose we pick a person at random from this sample. What is the probability that this person will be male       and has used cocaine at least 100 times?

Calculating Probabilities using the

classical approach with equally likely outcomes.

■   Toss a fair coin twice. What is the probability of observing exactly one head?

S = {HH, HT, TH, TT }

■   Now, what if we toss the coin 20 times? 

2.3 Counting Techniques

n !      

n Pr   = (n − r )!

■    Permutation with Repititions (indistinguishable objects): The

number of permutations of n objects of which n1 are alike, n2 are alike, …, nk are alike is

Example: How many different signals can be constructed from 4 identical blue flags and 2 identical red flags?

■    Combination: A selection of a subset of k items out of n taken

in such a way that the order of selection does not matter. The number of combinations of k items selected from n is given by

(n       P           n !      

■    Example: How many ways can we select a jury of 12 people from a group of 20?

Some Useful Results:

( n     ( n 

| 0)| = | n)| = 1

( n     ( n     

 = n

(n     (n               n !      

Examples: Counting TechniquesAt a certain café there are 6 choices of

bread, 5 choices of meat, 4 choices of cheese and 12 garnishes.  How

many sandwich possibilities are there if you choose

■   a) One bread, one meat and one cheese?

■   b) One bread, one meat, one cheese and from 0 to 12

garnishes?

■   c) One bread, 0,1, or 2 meats, 0,1 ,2 or 3 cheeses and

from 0 to 12 garnishes?

Example: A student takes a 20 question True/False test on which he guesses

a) How many ways can he answer the exam?

b) What is the probability he makes 100% on the exam?

c) What is the probability he makes exactly 80% on the exam?

d) What is the probability he passes the exam?

Suppose that 5 cards are selected without replacement

a) What is the probability of getting all cards of the same suit?

b) What is the probability of 2 Clubs and 3 Diamonds?

c) What is the probability of getting at least 4 clubs? 

If we select 5 cards with replacement from the deck, what is the

probability

a)    that all cards are the same suit?

b)     of getting 2 Clubs and 3 Diamonds?

c)     of getting at least 4 clubs?

2.4 Conditional Probability

■   Sometimes we are interested in restricting ourselves to a subset of the sample space instead of the entire sample space.

Def: The conditional probability of event A given that event B has

occurred is given by

Rule : The previous probability rules hold for conditional probabilities.

For example,

P (A ∪ C B ) = P (A B )+ P (C B ) − P ((A ∩ C ) B ) etc. 

Recall the study on cocaine by gender and frequency in The

American Journal of Drug and Alcohol Abuse with the following data.

Suppose we pick a person at     random from the  111 subjects and find that it is a male (M).    What is the probability that this male will be one who used          cocaine 100 times or more in   his life? 

Example :  For a certain married couple, the probability that the

husband will vote for a certain Presidential candidate is 0.6, while the probability that the wife will vote for the candidate is 0.4 and the probability they will both vote for the candidate is 0.2.

■   a) Find the probability that the wife votes for the candidate given that her husband votes for the candidate.

■   b) Find the probability that the wife votes for the candidate given that her husband does not vote for the candidate.

■   c) Find the probability that they both vote for the candidate given that at least one of them votes for the candidate.

■   Suppose that 50% of patients at a certain dentist office need x-

rays, and of those getting x-rays 20% will have a cavity that needs to be filled.  If a patient is selected at random, what is the           probability the patient will have x-rays and need to have a cavity  filled.

Remark 1: This can be illustrated with a tree diagram.

Remark 2: This can be extended to more than 2 events.

Example  1:  An  automobile  company  has  3  different  production

sites for  a  certain  car.    Four  percent  of  cars  from  site  1,  8%  of cars from site  2, and  6% of cars from site 3  have  been  recalled due to faulty air bags.  Suppose that 40% of cars are produced at site 1, 35% are produced at site 2, and 25% are produces at site 3.     If  a  car  of  this  type   is  selected  at   random,  what   is  the probability it is recalled for faulty air bags?

Referring to the last example, given that a randomly selected car has

been recalled for faulty air  bags, what  is the  probability  it came from site

2?

Example: Conditional Probability and Bayes Rule

A screening test indicates the presence or absence of a

particular disease or condition.

■   T+ denotes that an individual has a positive test result (indicating disease or condition)

■   T- denotes that an individual has a negative test result (indicating the disease or condition is not present)

■   D+ denotes that an individual actually has the disease or condition

■   D- denotes that an individual actually does not have the disease or condition

Terminology

Almost all of these tests have levels of error associated with their use.

■   False Positive:  the test indicates that a person has a condition when he or she does not

■   False Negative:  the test fails to show that a person has a disease or condition when he or she actually does have it

■   Sensitivity (SEN) is the probability that the test result will be  positive when the test is administered to people who actually have the disease or condition in question.

■   Specificity (SPEC) is the probability that a test will be negative when administered to people who are free of the disease or  condition in question. 

Example:   For Covid-19, the antibody test sold in the US by

Becton Dickinson, developed Biomedomics, has sensitivity of 88.7% and specificity of 90.6%. Suppose 15% of the            population being tested actually has Covid-19

■  Find the probability that a person will test negative.

■  Find the probability that someone who tests negative, actually does not have the disease.

2.5 Independence

Def:  Two events A and B are said to be independent if

otherwise they are said to be dependent.

Thm:  Two events A and B are said to be independent  if and only if

Def:  The events  are mutually independent if for each k

(k=2,3,…n) and every subset of indices

For example

Result: If the events A and B are independent then their

complements are also independent.

Example: It is known that a patient with a particular

disease will respond to treatment with probability

0.9.

■   If three patients are treated independently, what is the probability that all will respond?

■   At least one will respond?

Circuits or Systems in Series and Parallel: Let Ai be the event that

the ith component works.

■   Series:

■   Parallel:

■    Example : A system consists of 4 components connected as

shown below. If the components work or fail independently of one another and each works with probability 0.90, find the    probability the system works, event SW.

Example :  A system consists of components connected as shown

below.   If the components work or fail independently of one another with the given probabilities, find the probability the system works.