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# Category Archives: Discrete Random Variable

## Married Couples Expected Value & Variance example question

If 10 married couples are randomly seated at a round table, compute

(a) the Expected number

(b) the Variance

of the number of wives that are seated next to their husbands.

Solution to this Expected Value & Variance practice problem is given in the video below!

## Binomial Distribution Bayes Rule example question

A hospital receives of its flu vaccine shipments from Company X and the remainder of its shipments from other companies. Each shipment contains a very large number of vaccine vials. For Company X’s shipments, 10% of the vials are ineffective. For every other company, 2% of the vials are ineffective. The hospital tests 30 randomly selected vials from a shipment and finds that one vial is ineffective. What is the probability that this shipment came from Company X?

Solution to this Society of Actuaries Exam P practice problem is given in the video below!

## Binomial Distribution example problem

A company prices its hurricane insurance using the following assumptions:

(i) In any calendar year, there can be at most one hurricane

(ii) In any calendar year, the probability of a hurricane is 0.05

(iii) The number of hurricanes in any calendar year is independent of the number of hurricanes in any other calendar year.

Using the company’s assumptions, calculate the probability that there are fewer than 3 hurricanes in a 20-year period.

Solution to this Society of Actuaries Exam P practice problem is given in the video below!

## Binomial Distribution Hard example

A company establishes a fund of 120 from which it wants to pay an amount, C, to any of its 20 employees who achieve a high performance level during the coming year. Each employee has a 2% chance of achieving a high performance level during the coming year, independent of any other employee. Determine the maximum value of C for which the probability is less than 1% that the fund will be inadequate to cover all payments for high performance.

Solution to this Society of Actuaries Exam P practice problem is given in the video below!

## Binomial Distribution Multiple Random Variables example question

A study is being conducted in which the health of two independent groups of ten policyholders is being monitored over a one-year period of time. Individual participants in the study group drop out before the end of the study with probability 0.2 (independently of the other participants). What is the probability that at least 9 participants complete the study in one of the two groups, but not in both groups?

Solution to this Society of Actuaries Exam P practice problem is given in the video below!

## Poisson Distribution example problem

An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims. If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed?

Solution to this Society of Actuaries Exam P practice problem is given in the video below!

## Hypergeometric Distribution Probability example question

Out of 100 students qualifying for an exam, 20 were drawn randomly. If 65 out of 100 qualified students are male, what is the probability that 14 out of 20 chosen students are female?

Solution to this Hypergeometric Discrete Random Variable Distribution Probability practice problem is given in the video below!

## Negative Binomial Distribution Probability example question

An interviewer is given a list of potential people she can interview. If the interviewer needs to interview 5 people and if each person (independently) agrees to be interviewed with probability , what is the probability that her list of potential people will enable her to obtain her necessary number of interviews if the list consists of 5 people or 8 people?

Solution to this Negative Binomial Discrete Random Variable Distribution Probability practice problem is given in the video below!

## Poisson Distribution Probability example question

Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter = 3.

(a) Find the probability that 3 or more accidents occur today

(b) Repeat part (a) under the assumption that at least 1 accident occurs today

Solution to this Poisson Discrete Random Variable Distribution Probability practice problem is given in the video below!

## Geometric Distribution Probability example question

Consider a roulette wheel consisting of 38 numbers – 1 through 36, 0, and double 0. If Smith always bets that the outcome will be one of the numbers 1 through 12, what is the probability that Smith will lose his first 5 bets? What is the probability that his first win will occur on his fourth bet?

Solution to this Geometric Discrete Random Variable Distribution Probability practice problem is given in the video below!

## Binomial Distribution Probability example question

A multiple choice exam consists of 4 questions. Each question has 5 possible answers, exactly one of which is correct. Donald decides to guess all questions on the exam. What is the probability that he will answer exactly two questions correctly?

a)  0.1024

b)  0.9744

c)  0.2048

d)  0.1536

e)  0.0256

f)  none of the above

Solution to this Binomial Discrete Random Variable Distribution Probability practice problem is given in the video below!

## Hard Binomial Distribution Probability example problem

The efficacy of the mumps vaccine is about 80%; that is, 80% of those receiving the mumps vaccine will not contract the disease when exposed. Assume that each person’s response to the mumps is independent of another person’s response.

a) Find the probability that at least one exposed person will get the mumps if 10 people are exposed.

b) How many vaccinated people must be exposed to the mumps before the probability that at least one person will contract the disease is at least 0.95?

Solution to this Binomial Discrete Random Variable Distribution Probability practice problem is given in the video below!