1 General Probability

1.1 Supplementary Concepts

Axioms of probability

A probability function (\(\Pr\)) satisfies the following axioms:

Non-negativity:
For every event \(A\), \(\Pr[A]\geq 0\)
Normalization:
The probability of the sample space is \(1\), i.e., \(\Pr[\Omega]=1\)
Countable additivity:
If \(A_1,A_2,\ldots\) are mutually exclusive events, then \(\sum_{i=1}^{\infty}\Pr[A_i]=1\).
Events are mutually exclusive when no two can occur simultaneously, i.e., \(A_i\cap A_j=\varnothing\) for \(i\neq j\)

Permutations and combinations

A permutation is an ordered arrangement of objects. The number of ways to select and arrange \(r\) objects from \(n\) distinct objects is

\(_nP_r=\dfrac{n!}{(n-r)!}\)

A combination is an unordered selection of objects. The number of ways to select \(r\) objects from \(n\) distinct objects is

\(\dbinom{n}{r}=\dfrac{n!}{r!(n-r)!}\)

Conditional probability

Let \(A\) and \(B\) be events with \(\Pr[B]>0\). The conditional probability of \(A\), given that \(B\) has occurred, is

\(\Pr[A\mid B]=\dfrac{\Pr[A\cap B]}{\Pr[B]}\)
Conditioning on \(B\) restricts the sample space to outcomes in \(B\).

Multiplication rule

For three events, \(\Pr[A\cap B\cap C]=\Pr[A]\Pr[B\mid A]\Pr[C\mid A\cap B]\)

More generally, \(\Pr\left[\bigcap_{i=1}^{n}A_i\right]=\Pr[A_1]\prod_{i=2}^{n}\Pr[A_i\mid A_1\cap\cdots\cap A_{i-1}]\)

Independent events

Events \(A\) and \(B\) are independent if the occurrence of one does not change the probability of the other. Formally, \(\Pr[A\cap B]=\Pr[A]\Pr[B]\). When \(\Pr[B]>0\), independence is equivalent to \(\Pr[A\mid B]=\Pr[A]\). Similarly, when (>0), \(\Pr[B\mid A]=\Pr[B]\).

Events \(A_1,\ldots,A_n\) are mutually independent if every possible intersection satisfies the corresponding multiplication rule. For three events \(A\), \(B\), and \(C\), mutual independence requires \(\Pr[A\cap B]=\Pr[A]\Pr[B]\), \(\Pr[A\cap C]=\Pr[A]\Pr[C]\), \(\Pr[B\cap C]=\Pr[B]\Pr[C]\), and \(\Pr[A\cap B\cap C]=\Pr[A]\Pr[B]\Pr[C]\). Pairwise independence alone does not necessarily imply mutual independence.

Independence and mutual exclusivity are different concepts. If \(A\) and \(B\) are mutually exclusive, then \(\Pr[A\cap B]=0\). If \(A\) and \(B\) are independent, then \(\Pr[A\cap B]=\Pr[A]\Pr[B]\). If mutually exclusive events \(A\) and \(B\) both have positive probability, then \(\Pr[A\cap B]\neq\Pr[A]\Pr[B]\). Therefore,

\(\boxed{\text{Mutually exclusive events with positive probabilities are not independent}}\)

The occurrence of one mutually exclusive event makes the occurrence of the other impossible.

Law of total probability

Let \(B_1,\ldots,B_n\) form a partition of the sample space, with \(\Pr[B_i]>0\) for each \(i\). For any event \(A\),

\(\displaystyle \boxed{\Pr[A]=\sum_{i=1}^{n}\Pr[A\mid B_i]\Pr[B_i]}.\)

This formula separates the occurrence of \(A\) according to the mutually exclusive cases \(B_1,\ldots,B_n\).

For two complementary events \(B\) and \(B^c\),

\(\displaystyle \boxed{\Pr[A]=\Pr[A\mid B]\Pr[B]+\Pr[A\mid B^c]\Pr[B^c]}.\)

Bayes’ theorem

Let \(A\) and \(B\) be events with positive probability. By the multiplication rule,

\(\displaystyle \Pr[A\cap B]=\Pr[A\mid B]\Pr[B]\) and \(\displaystyle \Pr[A\cap B]=\Pr[B\mid A]\Pr[A]\)

Equating the two expressions gives \(\Pr[A\mid B]\Pr[B]=\Pr[B\mid A]\Pr[A].\) Therefore,

\(\displaystyle \boxed{\Pr[B\mid A]=\frac{\Pr[A\mid B]\Pr[B]}{\Pr[A]}}\)

If \(B_1,\ldots,B_n\) form a partition of the sample space, then

\(\displaystyle \boxed{\Pr[B_j\mid A]=\dfrac{\Pr[A\mid B_j]\Pr[B_j]}{\sum_{i=1}^{n}\Pr[A\mid B_i]\Pr[B_i]}}\)

Bayes’ theorem reverses the direction of conditioning.

Prior, likelihood, and posterior probabilities

In Bayes’ theorem, \(\Pr[B_j]\) is the prior probability of \(B_j\). The quantity \(\Pr[A\mid B_j]\) is the probability of observing \(A\) when \(B_j\) is true. It is often referred to as the likelihood of \(B_j\) given the observed evidence. The quantity \(\Pr[B_j\mid A]\) is the posterior probability of \(B_j\) after observing \(A\). Thus,

\(\displaystyle \boxed{\text{Posterior probability}=\frac{\text{Likelihood}\times\text{Prior probability}}{\text{Total probability of the evidence}}}\)

Summary of key probability rules

For any event \(A\), \(\Pr[A^c]=1-\Pr[A]\)
For any events \(A\) and \(B\), \(\Pr[A\cup B]=\Pr[A]+\Pr[B]-\Pr[A\cap B]\)
If \(A\) and \(B\) are mutually exclusive, \(\Pr[A\cup B]=\Pr[A]+\Pr[B]\)
If \(\Pr[B]>0\),
\(\displaystyle \Pr[A\mid B]=\frac{\Pr[A\cap B]}{\Pr[B]}\)
For any events \(A\) and \(B\), \(\Pr[A\cap B]=\Pr[A]\Pr[B\mid A]\)
If \(A\) and \(B\) are independent, \(\Pr[A\cap B]=\Pr[A]\Pr[B]\)
If \(B_1,\ldots,B_n\) form a partition of the sample space, \(\displaystyle \Pr[A]=\sum_{i=1}^{n}\Pr[A\mid B_i]\Pr[B_i]\)
Bayes’ theorem is \(\displaystyle \Pr[B_j\mid A]=\dfrac{\Pr[A\mid B_j]\Pr[B_j]}{\sum_{i=1}^{n}\Pr[A\mid B_i]\Pr[B_i]}\)

1.2 Supplementary Exercises

If four actuaries, three accountants, and three economists are to be seated in a row, how many seating arrangements are possible when people of the same profession must sit next to one another?

An actuarial student is to answer seven out of \(10\) questions in an examination.

How many choices has she?
How many if she must answer at least three of the first five questions?

How many seven-character passwords are possible when three of the characters must be letters (set of \(26\)) and four of the characters must be digits?

Consider three classes, each consisting of \(10\) students.

From this group of \(30\) students, a group of three students is to be chosen to speak at a conference.

How many choices are possible?
How many choices are there in which all three students are from the same class?
How many choices are there in which two of the three students are in the same class and the other student is in a different class?
how many choices are there in which all three students are in different classes?

A committee of six people is to be chosen from a group of seven men and eight women.

If the committee must consist of at least three women and at least two men, how many different committees are possible?

How many subsets of size four of the set \(S={1, 2, …, 20}\) contain at least one of the elements \(1\), \(2\), \(3\), \(4\), \(5\)?

In how many ways can three calculus books, two algebra books, and one trigonometry book be arranged on a bookshelf if:

the books can be arranged in any order?
the calculus books must be together and the algebra books must be together?
the calculus books must be together, but the other books can be arranged in any order?

A team of nine, consisting of two mathematicians, three statisticians, and four actuaries, is to be selected from a faculty \(10\) mathematicians, eight statisticians, and seven actuaries.

How many teams are possible?

An actuarial student has eight friends, of whom five will be invited to a party.

How many choices are there if two of the friends are feuding and will not attend together?
How many choices if two of the friends will only attend together?

Twenty items are arranged in a five-by-four array as shown.

\(A_{1}\)	\(A_{2}\)	\(A_{3}\)	\(A_{4}\)	\(A_{5}\)
\(A_{6}\)	\(A_{7}\)	\(A_{8}\)	\(A_{9}\)	\(A_{10}\)
\(A_{11}\)	\(A_{12}\)	\(A_{13}\)	\(A_{14}\)	\(A_{15}\)
\(A_{16}\)	\(A_{17}\)	\(A_{18}\)	\(A_{19}\)	\(A_{20}\)

Calculate the number of ways to form a set of three distinct items such that no two of the selected items are in the same row or same column.

A pair of dice is tossed and the two numbers appearing on the top are recorded.

Find the number of elements in each of the following events:

\(A=\{\)two numbers are equal\(\}\)
\(B=\{\)sum is \(\geq 10\) \(\}\)
\(C=\{5\) appears on first die\(\}\)
\(D=\{5\) appears on at least one die\(\}\)

A box contains \(15\) billiard balls numbered \(1\) to \(15\).

A ball is drawn at random and the number recorded.

Find the probability \(p\) that the number is:

even
less than \(5\)
even and less than \(5\)
even or less than \(5\)

Six married couples are standing in a room. Two people are chosen at random.

Find the probability \(p\) that:

they are married
one is male and one is female

A university cafeteria offers a three-course meal consisting of an entrée, a main course, and a dessert. The possible choices are:

Course	Choices
Entrée	Tacos or quesadillas
Main course	Burrito or enchiladas or chilaquiles
Dessert	Arroz con leche or flan or dulce de leche

A person is to choose one course from each category.

How many outcomes are in the sample space?
Let \(A\) be the event that arroz con leche is chosen. How many outcomes are in \(A\)?
Let \(𝐵\) be the event that tacos are chosen. How many outcomes are in \(B\)?
List all the outcomes in the event \(A\cap B\)
Let \(C\) be the event that enchiladas are chosen. How many outcomes are in \(C\)?
List all the outcomes in the event \(A\cap B\cap C\)

Urn \(A\) contains three red and three white balls, whereas urn \(B\) contains four red and six white balls.

If a ball is randomly selected from each urn, what is the probability that the two balls will be the same color?

Of \(12\) actuaries applying for a job, eight have work experience, six have passed at least one SOA exam, and four have work experience but have not passed any SOA exam.

The hiring company randomly selects a candidate for an interview.

What is the probability that the selected candidate has work experience and has passed at least one SOA exam?

An insurance company estimates that \(40\%\) of policyholders who have only an auto policy will renew next year and \(60\%\) of policyholders who have only a homeowners’ policy will renew next year.

The company estimates that \(80\%\) of policyholders who have both an auto policy and a homeowners’ policy will renew at least one of those policies next year.

Company records show that \(65\%\) of policyholders have an auto policy, \(50\%\) of policyholders have a homeowners’ policy, and \(15\%\) of policyholders have both an auto policy and a homeowners’ policy.

Using the company’s estimates, calculate the percentage of policyholders that will renew at least one policy next year.

You are given \(Pr[A \cup B]=0.7\) and \(Pr[A \cup B \,']=0.9\).

Calculate \(Pr[A]\).

An insurance company examines its pool of auto insurance customers and gathers the following information:

All customers insure at least one car
\(70\%\) of the customers insure more than one car
\(20\%\) of the customers insure a sports car
Of those customers who insure more than one car, \(15\%\) insure a sports car

Calculate the probability that a randomly selected customer insures exactly one car, and that the car is not a sports car.

A survey of a group’s viewing habits over the last year revealed the following information:

\(28\%\) watched gymnastics
\(29\%\) watched baseball
\(19\%\) watched soccer
\(14\%\) watched gymnastics and baseball
\(12\%\) watched baseball and soccer
\(10\%\) watched gymnastics and soccer
\(8\%\) watched all three sports

Calculate the percentage of the group that watched none of the three sports during the last year.

Charlie is undecided as to whether take exam FM or exam P. He estimates that his probability of passing FM is \(0.6\), and his probability of passing P is \(0.7\).

If Charlie is to base his decision on the flip of a fair coin, what is the probability that he passes the P exam?

An insurance company believes that people can be divided into two classes: those who are accident-prone and those who are not.

The company’s statistics show that an accident-prone person will have an accident at some time within a fixed one-year period with probability \(0.4\), whereas this probability decreases to \(0.2\) for a person who is not accident prone.

If is is assumed that \(30\%\) of the population is accident-prone, what is the probability that a new policyholder will have an accident within a year of purchasing a policy?

Consider three eveents: \(A\), \(B\), and \(C\). Find \(Pr[A]\) given that:

\(A\) and \(C\) are independent
\(B\) and \(C\) are independent
\(A\) and \(B\) are disjoint
\(Pr[A \cup B]=\frac{2}{3}\), \(Pr[B \cup C]=\frac{3}{4}\), and \(Pr[A \cup B \cup C]=\frac{11}{12}\)

In a certain city, it rains one-third of the days.

Given that it is rainy, there will be heavy traffic with probability \(0.5\), and given that it is not rainy, there will be heavy traffic with probability \(0.25\).

If it is rainy and there is heavy traffic, Charlie arrives late for class with probability \(0.5\).

But the probability of being late is reduced to \(0.125\) if it is not rainy and there is no heavy traffic.

In other situations (rainy and no traffic, not rainy and traffic) the probability of Charlie being late is \(0.25\).

Assuming that a random day is picked,

What is the probability that it is not raining and there is heavy traffic, and Charlie is not late for class?
What is the probability that Charlie is late for class?
Given that Charlie arrived late for class, what is the probability that it rained that day?

Consider three bags, each containing \(100\) balls:

Bag \(A\) has \(75\) orange and \(25\) white balls
Bag \(B\) has \(60\) orange and \(40\) white balls
Bag \(C\) has \(45\) orange and \(55\) white balls

One bag is chosen at random, and a ball is picked.

What is the probability that the picked ball is orange?

A certain disease affects one in \(10\,000\) people.

There is a test to check whether the person has the disease. The test is quite accurate, however,

The probability that the test result is positive (i.e., the person has the disease) given that the person does not have the disease is \(2\%\)
The probability that the test result is negative given that the person has the disease is \(1\%\)

A random person is tested, and the result is positive.

What is the probability that the person has the disease?

An actuarial student will watch two movies in a weekend.

Suppose that the probability that likes movie \(B\) is \(0.6\), the probability that she will like movie \(O\) is \(0.5\), and the probability that she will like the two movies is \(0.4\).

Find the conditional probability that she will like movie \(O\) given that she did not like movie \(B\).

Two factories, \(A\) and \(B\), produce smartphones.

Each smartphone produced at factory \(A\) is defective with probability \(0.05\), whereas each one produced by factory \(B\) is defective with probability \(0.01\).

You purchased two smartphones that were produced by the same factory, which is equally likely to have been either factory \(A\) or factory \(B\).

If the first smartphone that you check is defective, what is the conditional probability that the other one is also defective?

If four balls are randomly chosen from an urn containing four red, five white, six blue, and seven green balls, find the conditional probability that they are all white given that all balls are of the same color.

A researcher examines medical records of \(937\) men who died in 2022 and discovers that \(210\) of the men died from heart disease.

Moreover, \(312\) of the \(937\) men had at least one parent who suffered from heart disease, and of the \(312\) men, \(102\) died from heart disease.

Determine the probability that a man randomly selected from this group died from heart disease, given that neither of his parents suffered from heart disease.

Upon arrival at a hospital, patients are categorized according to condition: critical, serious, or stable.

In 2025:

Ten percent of patients were critical;
Thirty percent of patients were serious;
The rest of patients were stable;
Forty percent of the critical patients died;
Ten percent of the serious patients died; and
One percent of the stable patients died.

Given that a patient survived, what is the probability that the patient was categorized as serious upon arrival?

For a certain population, \(0.8\), \(0.4\), and \(0.1\) are the probabilities that a newborn survives to ages \(55\), \(75\), and \(85\), respectively.

What is the probability that a person aged \(55\):

survives to age \(75\)?
survives to age \(85\)?