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# Category Archives: Binomial Distribution

## Normal Approximation for Binomial Random Variable example problem

The median age of residents of the United States is 37.2 years. If a survey of 200 residents is taken, approximate the probability that at least 110 will be under 37.2 years of age.

Solution to this probability Binomial Distribution Normal Approximation practice problem is given in the video below!

## Binomial Random Variable Normal Approximation HARD example question

The weekly amount sent by a small company for in-state travel has approximately a normal distribution with mean \$1450 and standard deviation \$220.

(a) What is the probability that the actual expenses will exceed \$1560 in 20 or more weeks during the next year?

(b) What is the probability that the actual expenses would exceed \$1500 for between 18 and 24 weeks, inclusive during the next year?

Solution to this probability Binomial Distribution Normal Approximation 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!

## Expected Value example question

A box contains 8 green and 4 blue marbles. Two marbles are selected at once without replacement. What is the expected number of green marbles among the selected ones?

a)  1

b) c) d) e) f)  none of the above

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

## Expected Value example problem #2

You have 5 pairs of shoes. Four of them are worth \$30 each, while the fifth is worth \$2,000. You select a pair at random. What is the expected value of the pair you have selected?

a)  \$424

b)  \$30

c)  \$50.20

d)  \$432

e)  \$1,030

f)  none of the above

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

## Binomial Distribution Expectation example

A sample of 3 items is selected at random from a box containing 20 items of which 4 are defective. Find the expected number of defective items in the sample.

Solution to this Binomial Random Variable Expected Value practice problem is given in the video below!

## Hypergeometric Distribution Expectation example question

A ball is chosen at random from each of 5 urns. Each urn contains balls as follows:

urn 1: 1 white, 5 black

urn 2: 3 white, 3 black

urn 3: 6 white, 4 black

urn 4: 2 white, 6 black

urn 5: 3 white, 7 black

Compute the expected number of white balls selected.

Solution to this Hypergeometric Random Variable Expected Value practice problem is given in the video below!

## Continuous Distribution Expected Value example problem

The density function of X is given by

f(x) = { 0 otherwise

If E[X] = , find the values of constants and .

Solution to this Continuous Random Variable Expected Value practice problem is given in the video below!

## Independent & Identically Distributed UNIFORM Random Variables Expected Value example

If X1, X2, …, Xn are independent and identically distributed random variables having uniform distributions over (0,1),

find

E [max(X1, …, Xn)]

and

E [min(X1, …,Xn)]

Solution to this Uniform Random Variable Expected Value 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!