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Conditional Expectation problems

Geometric Distribution Conditional Expectation Probability example question

A fair die is successively rolled. Let X and Y denote, respectively, the number of rolls necessary to obtain a 6 and a 5.

Find

E [X] ,

E [X \dpi{150} \large | Y = 1]

and

E [X \dpi{150} \large | Y = 5] .

 

Solution to this Conditional Expectation Probability practice problem is given in the video below!



 

Joint Moment Generating Function problems

Joint Moment Generating Function example question

The joint density function of X and Y is given by

f(x, y) = 

Find the Joint Moment Generating Function of X and Y.

 

Solution to this Joint Moment Generating Function practice problem is given in the video below!



Expected Value problems

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!



Joint Distribution Probability Density Function problems

Joint Probability Density Function example question

The joint probability density function of X and Y is given by

f(x,y) =

a. Find the density function of X

b. Find P(X > Y)

c. Find P(Y >  \dpi{150} \LARGE | X <  )

 

Solution to this Joint Probability Density Functions practice problem is given in the video below!



Exponential Distribution Continuous Random Variable problems

Exponential Distribution Probability example question

The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter =  . What is the probability that a repair time exceeds 2 hours? What is the conditional probability that a repair takes at least 10 hours, given that its duration exceeds 9 hours?

 

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



Normal Distribution Continuous Random Variable problems

Normal Distribution Probability example question

The annual rainfall (in inches) in a certain region is normally distributed with = 40,  = 4. What is the probability that starting with this year, it will take over 10 years before a year occurs having a rainfall of over 50 inches?

 

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



Normal Distribution PERCENTILE Probability example problem

The measure of intelligence of a group of people is assumed to be approximately normally distributed with mean 102 and standard deviation 14. If we are told that Tom’s IQ denoted by xT is at the 98th percentile of the IQ distribution of the group of people, find xT.

 

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



Uniform Distribution Continuous Random Variable problems

Uniform Distribution Probability example question

You arrive at a bus stop at 10 o’clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30. What is the probability that you will have to wait longer than 10 minutes? If at 10:15 the bus has not yet arrived, what is the probability that you will have to wait at least an additional 10 minutes?

 

Solution to this Uniform Continuous Random Variable Distribution Probability 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!



Continuous Distribution Random Variable problems

Continuous Distribution Probability example question

X is a continuous random variable with probability density function given by

f(x) = {

0 otherwise

a. Find the value of C

b. What is the probability that X > 1?

 

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



Hypergeometric Distribution Discrete Random Variable problems

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 Discrete Random Variable problems

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 Discrete Random Variable problems

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 Discrete Random Variable problems

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 Discrete Random Variable problems

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!



 

Independent Events Probability problems

Independent Events Probability example question

Two independent events have P(A) = P(B) = 0.3. What is the P(A  B)?

a)  0.09

b)  0.91

c)  0.49

d)  0.51

e)  0.6

f)  none of the above

 

Solution to this Independent Events Probability practice problem is given in the video below!



Independent Events Probability example problem

For two independent events we know that P(A) = 0.7 and P(A  B) = 0.21. What is P(B)?

a)  0.7

b)  0.6

c)  0.5

d)  0.4

e)  0.3

f)  none of the above

 

Solution to this Independent Events Probability practice problem is given in the video below!



Independent Events Probability example

A car of brand X has a transmission which fails with probability 0.6 and brakes which fail with probability 0.3; the two kinds of failures occur independently. At any given day, what is the probability that exactly one of the failures occurs while driving the car of brand X?

a)  0.72

b)  0.28

c)  0.46

d)  0.54

e)  0.90

f)  none of the above

 

Solution to this Independent Events Probability practice problem is given in the video below!



Venn Diagram and Conditional Probability problems

Venn Diagram Bayes Rule Probability example question

For two events we have P(A) = 0.29, P(B) = 0.43, and P(A B) = 0.65. What is P(A  B)?

a)  0.27

b)  0.07

c)  0.16

d)  0.43

e)  0.08

f)  none of the above

 

Solution to this Venn Diagram Probability practice problem is given in the video below!



Venn Diagram Bayes Rule Probability example problem #2

A universal set U = {1, 2, 3, 4, 5, 6, 7, 8} has subsets A = {1, 2, 3, 4} and B = {1, 2, 6, 7}. What set is A’  B?

a) 

b)  {1, 2, 5, 6, 7, 8}

c)  {1, 2, 3, 4}

d)  {1, 2, 3, 4, 5, 8}

e)  {1, 2}

f)  none of the above

 

Solution to this Venn Diagram Probability practice problem is given in the video below!



Venn Diagram Probability word problem example

A manufactured component has its quality graded on its performance, appearance, and cost. Each of those three characteristics is graded as either pass or fail. There is a probability of 0.40 that a component passes on both appearance and cost. There is a probability of 0.31 that a component passes on all three characteristics. There is a probability of 0.64 that a component passes on performance. There is a probability of 0.19 that a component fails on all three characteristics. There is a probability of 0.06 that a component passes on appearance but fails on both performance and cost.

a) What is the probability that a component passes on cost but fails on both performance and appearance?

b) If a component passes on both appearance and cost, what is the probability that it passes on all three characteristics?

 

Solution to this Venn Diagram Probability practice problem is given in the video below!



 

 

Venn Diagram & Bayes Rule Conditional Probability example question #3

How does VENN Diagram relate to Bayes Formula

when finding Conditional Probability?

 

Solution to this Venn Diagram & Bayes Formula Conditional Probability practice problem is given in the video below!



 

 

Conditional Probability Bayes Rule example problem #4

On Tuesday morning, David randomly picks a microphone, and it fails. What is the probability that the microphone of brand Y was chosen, given that probability of failure for microphone X is 0.3, probability of failure for microphone Y is 0.4, probability of choosing microphone X is , and probability of choosing microphone X is  ?

a) 

b) 

c) 

d) 

e) 

f)  none of the above

 

Solution to this Conditional Probability Bayes practice problem is given in the video below!



 

 

Conditional Probability Bayes Rule example #5

A car brand X produces 40% of its cars at plant A and the remainder at plant B. Of all cars produced at plant A, 20% do not have a spare tire, while 30% of the cars produced at B do not have a spare tire. Car X is purchased, and it happens to have a spare tire. What is the probability that it was produced at plant B?

a) 

b) 

c) 

d)  0.7

e)  0.8

f)  none of the above

 

Solution to this Conditional Probability Bayes practice problem is given in the video below!



 

 

Conditional Probability Bayes Rule example question #6

David has two microphones which he uses to teach his Algebra class; one is brand X and the other is brand Y. Microphone X fails with probability 0.3, while brand Y fails with probability 0.4. On a particular morning, David picks a microphone at random. What is the probability that it will fail?

a)  0.7

b)  0.35

c)  0.12

d)  0.58

e)  0.42

f)  none of the above

 

Solution to this Conditional Probability Bayes practice problem is given in the video below!