Probability models
01
Venn diagrams - set rules and Kolmogorov additivity
Suppose we know three probabilities of events:
, , and . Calculate:
, , , , and . Link to originalSolution
04
(1)
is computed by directly applying the inclusion-exclusion principle.
(2)
(3)
(4) We can express
as . Therefore,
(5)
Link to original
02
Inclusion-exclusion reasoning
Your friend says: “according to my calculations, the probability of
is and the probability of is , but the probability of and both happening is only .” You tell your friend they don’t understand probability. Why?
Link to originalSolution
05
(1) State the inclusion-exclusion principle.
(2) Examine possibilities based on given values.
Given that
is and is , we have that . Since
, we know that . Therefore,
Link to original.
04
At least two heads from three flips
A coin is flipped three times.
What is the probability that at least two heads appear?
Link to originalSolution
07
(1) Describe the sample size of this experiment.
(2) Find the probability that at least two heads appear.
The sequences of flips that contain at least two heads are
, , , . We know that
, thus Link to original
Conditional probability
02
Conditioning - two dice, at least one is 5
Two dice are rolled, and at least one is a 5.
What is the probability that their sum is 10?
Link to originalSolution
09
(1) Let
be the outcome of the first die and be the outcome of the second die. We are asked to compute
(2) Compute individual probabilities.
There is only one combination out of the 36 possible combinations of two dice rolls in which at least 1 die rolls a 5 and both sum up to 10 (5, 5).
There are
combinations of dice rolls in which at least one is a .
(3) Plug into formula.
Link to original
04
Multiplication - drawing two hearts
Two cards are drawn from a standard deck (without replacement).
(a) What is the probability that both are hearts?
(b) What is the probability that both are 4?
Link to originalSolution
16
(a)
Let
be the outcome of the first card, be the outcome of the second card, and denote “hearts”. Since there are 52 cards in a standard deck with 13 of them being hearts, we have (b)
Similarly to part (a), we have
Link to original
06
Conditioning relation
Suppose you know
and and . Calculate
and and . Link to originalSolution
15
(1) Set up conditional probability formula.
Solve for
.
(2) Plug in given values.
(3) Set up conditional probability formula.
Solve for
.
(4) Plug in given values.
(5) Use inclusion-exclusion principle to find
. Link to original
Bayes’ Theorem
02
Bayes’ Theorem - Inferring die from roll
A bag contains one 4-sided die, one 6-sided die, and one 12-sided die. You draw a random die from the bag, roll it, and get a 4.
What is the probability that you drew the 6-sided die?
Link to originalSolution
05
(1) Define events.
Let
be the event in which you draw the 4-sided die, be the event in which you draw the 6-sided die, and be the event in which you draw the 12-sided die. We are asked to compute
.
(2) Define obvious probabilities.
.
.
.
.
(3) Use Bayes’ Theorem to set up the formula for
.
(4) Plug in values and solve.
Link to original
03
Bayes’ Theorem - DNA evidence
A crime is committed in a town of 100,000 citizens. After all 100,000 citizens’ DNA is analyzed, your friend Jim is found to have a DNA match to evidence at the scene. A forensics expert says that the probability of a random person matching this evidence is 0.01%. How likely is it that Jim is guilty?
Link to originalSolution
01
(1) Define events.
Let
be the event that Jim is guilty. Let
be the event in which the DNA matches. We are given that
. We know that
. Since there are 100,000 citizens,
, . We are asked to compute
.
(2) Use Bayes’ Theorem to set up the formula for
.
(3) Plug in values.
Link to original
Independence
03
Applicant qualifications B
A hiring manager will randomly select two people from a group of 5 applicants. Of the 5 applicants, 2 are more qualified and 3 are less qualified (but the manager does not know this).
Let Event A be selecting 2 more qualified applicants and Event B be selecting 2 less qualified applicants. Determine whether A and B are independent events and justify your answer.
Link to originalSolution
Tree diagrams
01
Bin of marbles
A bin contains 5 red marbles, 7 blue marbles, and 3 white marbles.
We draw a random marble. If it’s red, we put it back, if it’s not red, we keep it. We do this three times.
(a) What is the probability of getting red then white then blue?
(b) Suppose the last draw was blue. What is the probability that the first was red?
Link to originalSolution
09
(a)
(1) Define events.
We define the sample space as
.
is the first ball drawn, the second, and the third.
represents the event in which a red ball is drawn, represents the event in which a white ball is drawn, represents the event in which a blue ball is drawn.
(2) Compute
.
(b)
(1) We are asked to compute
. Use Bayes’ Theorem to set up formula for .
(2) Find probabilities for all relevant combinations.
(3) Plug in values.
Link to original
Counting
01
Counting outcomes - permutations and combinations
In a lottery, five distinct numbers are picked at random from
. How many possible outcomes are there: (a) If we care about the order of numbers.
(b) If the order does not matter.
Link to originalSolution
10
(a)
If the order matters, then we are dealing with a permutation. We want
distinct numbers, so
(b)
If the order matters, then we are dealing with a combination. We want
distinct numbers, so Link to original
03
Drawing balls of distinct color
A bin contains 3 green and 4 yellow balls. Two balls are drawn out.
What is the probability that they are different colors?
Link to originalSolution
12
(1) Define relevant events.
If two balls of different colors are drawn out, then we choose one of each color.
Since order doesn’t matter, our sample space consists of all the ways we choose 2 balls out of 7.
(2) Compute probability.
Link to original
05
Binomial - Repeated coin flips
A coin is flipped 7 times and the sequence of results recorded as an outcome.
(a) How many possible outcomes have exactly 3 heads?
(b) How many possible outcomes have at least 3 heads?
Link to originalSolution
12
(a)
Out of
trials, we choose of them to be heads. Thus,
(b)
Out of 7 trials, we choose at least 3 of them to be heads.
Using summation notation, we get
Link to original
08
Counting license plates
A license plate must consist of 3 letters followed by 4 digits. Assuming we choose from 26 letters, A-Z, and 10 digits, 0-9, how many different license plates could be created if:
(a) Letters and numbers cannot be repeated.
(b) Letters and numbers can be repeated except that there must be exactly two 9’s.
Link to originalSolution
06
(a) Letters in
ways Digits in ways For each 3 letter arrangement, there are 5040 digit arrangements. Total number of ways (b) Letters in
ways Decide the spots for the two 9’s first: Remaining 2 digits in ways Total number of ways: Link to original
Repeated trials
01
Independent trials - At least 45 good paper clips
For a paper clip production line, 90% of the paper clips come off good, and 10% come off broken.
You buy a box of 50 paper clips from this line. What is the probability that at least 45 of them are good?
Link to originalSolution
05
(1) State the formula for a binomial distribution.
(2) State parameters of binomial distribution.
since you buy paper clips.
ranges from to .
.
.
(3) Use summation notation to find
. Link to original
04
Guessing on a test
Your odds of getting any given exam question right are
. The exam has 4 questions, and you need to answer 3 correctly to pass. (a) What is the probability that you pass?
(b) After finishing the exam, you are 100% sure that you got the second question right. Now what are the odds that you pass?
Link to originalSolution
08
(1) The probability that you pass follows a binomial distribution.
since there are 4 questions to account for. We choose
questions to be correct, where or .
(2) Describe the situation wherein you get the second question right.
Now, out of the remaining three questions, you need to answer at least two correctly.
This too follows a binomial distribution.
(3) Set up binomial distribution formula.
, since we only need to account for the remaining 3 questions. We choose
questions to be correct, where or . Link to original
06
Watching the Superbowl
A representative from Nielsen ratings randomly selects people in Charlottesville, VA and asks them whether they watched the Superbowl. The probability that any individual in Charlottesville watched the Superbowl is 0.3.
(a) What is the probability that if the representative asks 10 people, that less than 2 of them will have watched the Superbowl?
(b) What is the probability that the representative will have to ask at least 3 people to find someone who watched the Superbowl?
Link to originalSolution
13
(a)
(b)
Link to original
Reliability
01
Reliability for complex process
Consider a process with the following diagram of components in series and parallel:
Use
to denote the event that component succeeds. Suppose the success probabilities per component are given by this chart:
1 2 3 4 5 6 7 8 80% 60% 40% 90% 80% 50% 70% 90% What are the odds of success for the whole process?
Link to originalSolution
09
(1) Describe dependencies of components.
The series 4, 5 runs parallel to component 3.
The components 2, 3, 4, 5, 6 all run in a series.
The latter is parallel to component 7
The circuit starts at component 1 and ends at component 8
(2) Find the probability the component 4 and 5 succeeded. (Denote as
)
(3) Find the probability 3, and
succeed. Note that these are in parallel. (Denote as )
(4) Find the probability 2,
, and 6 succeed. (Denote as )
(5) Find the probability 7 and
succeed. Note that these are in parallel. (Denote as )
(6) Lastly, find the probability
, , and 8 succeed. (Denote as ). Link to original
04
Reliability - Math competition cutoff score
At a high school math competition, students take a test with 10 questions. Each question is worth one point and the probability of a student getting any one question correct is 0.55, independent of the other questions.
(a) Find the probability of a student getting a score of 8 or higher.
(b) Students take the test individually but compete in teams of 2. To proceed to the second round of competition, each student on the team must score at least 8. Each high school can enter 2 teams. If a high school enters two teams, find the probability at least one of their teams will make it to the second round. Assume students’ scores are independent.
Link to originalSolution
16
(a)
(b)
Link to original
Discrete random variables
06
Variance from CDF: Drill bit changes
The bits for a particular kind of drill must be changed fairly often. Let
denote the number of holes that can be drilled with one bit. The CDF of is given below: (a) Find the probability that a bit will be able to drill more than 2 holes.
(b) Find
by constructing the PMF. Link to originalSolution
10
(a)
(b)
Link to original
Bernoulli process
03
Rolling until a six
A fair die is rolled until a six comes up.
What are the odds that it takes at least 10 rolls? (Use a geometric random variable.)
Link to originalSolution
13
(1) Define random variables.
Let
. We wish to find
. For all
, , the first trials result in failure, and the trial is a success.
(2) Compute probability.
Note that the summation is simplified using the formula for a geometric series.
Link to original
04
Intersection accidents
Suppose that the odds of an accident occurring on any given day at the intersection of Ivy and Emmet is 0.05.
What are the odds of the first accident occurring between day 5 and day 10, inclusive? (Use an appropriate discrete distribution type.)
Link to originalSolution
04
(1) Define random variables.
Let
. We wish to find
.
(2) Compute
using the formula for a geometric distribution. Link to original
08
Prize on the Mall
A booth on the Mall is running a secret prize game, in which the
passerby wearing a hat wins $1,000. Passersby wear hats independently of each other and with probability 20%.
Let
be a random variable counting how many passersby pass by before a winner is found. (a) What is the name of the distribution for
? What are the parameters? (b) What is the probability that the
passerby wins the prize? (c) What is the probability that at least
passersby are needed before a winner is found? Link to originalSolution
08
(a)
Identify the distribution.
follows a negative binomial distribution with parameters and .
(b)
Compute
.
(c)
Compute
. We know that the minimum number of passersby before a winner is declared is
. Therefore,
. Link to original
Expectation and variance
02
Insurance expected payout
A car insurance analytics team estimates that the cost of repairs per accident is uniformly distributed between $100 and $1500. The manager wants to offer customers a policy that has a $500 deductible and covers all costs above the deductible.
How much is the expected payout per accident?
(Hint: Graph the PDF for the cost of repairs
; write a formula for the payout in terms of using cases; then integrate.) Link to originalSolution
06
(1) Find PDF of
. If
, insurance covers 0$. If
, then the insurance covers dollars.
(2) Integrate to find
. Since the cost of repairs in uniformly distributed, we have
, . Link to original
04
Tutoring needs
A course with 6 students offers free one-on-one tutoring to each student for 1 hour the week before the final exam. One tutor, Jim, has been hired to provide this tutoring, but he is available for only 4 hours that week. The instructor of the course will tutor any students that Jim is not able to help. Jim will be paid $20 per hour by the department. The instructor will provide tutoring for free. Let
be the number of students that will need tutoring. The PMF of is given below. (a) Find the probability the instructor will need to provide tutoring.
(b) Find the expected value of the number of students that will need tutoring.
Link to originalSolution
13
(a)
(b)
Link to original
05
Expectation from CDF
The CDF of random variable X is given by:
Compute
. Link to originalSolution
16
(a)
(b)
Link to original
Poisson process
03
Application of Poisson: meteor shower
The UVA astronomy club is watching a meteor shower. Meteors appear at an average rate of
per hour. (a) Write a short explanation to justify the use of a Poisson distribution to model the appearance of meteors. Why should appearances be Poisson distributed?
(b) What is the probability that the club sees more than 2 meteors in a single hour?
(c) Suppose we learn that over a four hour evening, 13 meteors were spotted. What is the probability that none of them happened in the first hour?
Link to originalSolution
03
(a)
Write explanation.
- You use Poisson distribution if events occur randomly and you know the mean number of events that occur within a given interval of time.
- In addition, Poisson distributions are advantageous when describing rare events. Since meteors are a rare occurrence, it makes sense to use a Poisson distribution.
(b)
Compute probability.
Since
, it’s easier to compute the latter.
(c)
Compute probability.
We know that there are 13 meteors in 4 hours, so we see an average of
meteors per hour. Let We wish to find the probability
. Link to original
Function on a random variable
01
Constants in PDF from expectation
Suppose
has PDF given by: Suppose
. Find the only possible values for and . Then find . Link to originalSolution
06
(1) Recall formula for expectation of a continuous random variable.
(2) Use formula to find an equation relating
and .
(3) Recall that integrating a PDF should yield
. Integrate the PDF and find a second equation relating and .
(4) Solve system of equations for
and . Isolating
in the second equation yields . Plugging this expression into the first equation yields
. Solving for
yields 2.4, and thus .
(5) Compute variance using the formula
Compute
. Link to original
03
PDF of derived variable for
and Suppose the PDF of an RV is given by:
(a) Find
using the integral formula. (b) Find
, the PDF of (by calculating the CDF first). (c) Find
using . (d) Find
using results of (a) and (c). Link to originalSolution
08
(a)
Compute
.
(b)
(1) Find the CDF of
.
(2) Find the CDF of
. Since
is monotone increasing, .
(3) Find the PDF of
by differentiating.
(c)
Find
.
(d)
Compute
. Link to original
05
CDF of derived variable
Suppose
is a continuous random variable. Let . Find the CDF of . Link to originalSolution
09
Case 1:
Case 2:
Therefore:
Link to original
Continuous wait times
04
Selling Christmas trees
An online company sells artificial Christmas trees. During the holiday season, the amount of time between sales,
, is an exponential random variable with an expected value of 2.5 hours. (a) Find the probability that the store will sell more than 2 trees in a 1-hour period of time.
(b) Find the probability the time between the sales of two trees will be between 4-5 hours.
Link to originalSolution
15
(a)
(b)
Link to original
05
Potholes on the highway
On a certain terrible stretch of highway, the appearance of potholes can be modeled by a Poisson process. Let the RV
denote the distance between successive potholes (measured in miles). The CDF of X is: (a) What is the mean number of potholes in a 2-mile stretch of the highway?
(b) What is the probability that there will be at least 2 potholes in a 2-mile stretch of the highway?
Link to originalSolution
14
(a)
(b)
Link to original
Normal distribution
03
Generalized normal
Let
be generalized normal variable with and . Using a chart of values, find: (a)
(b)
such that (c)
(Hint: Use and to avoid integration.) Link to originalSolution
03
(a)
(1) Write desired probability in terms of
values.
(2) Evaluate using a
value table.
(b)
(1) Use the
value table to find if
(2) Solve for
knowing that .
(c)
Recall formula for
and solve for Plug in
for and for and compute . Link to original
04
Normal distribution - test scores
In a large probability theory exam, the scores are normally distributed with a mean of 75 and a standard deviation of 10.
(a) What is the probability that a student scored between 70 and 80
(b) What is the lowest score a student can achieve to be in the top 5%?
(c) What score corresponds to the 25th percentile?
Link to originalSolution
04
(a)
(1) Write desired probability in terms of
values.
(2) Use table to evaluate expression.
(b)
(1) Interpret problem.
Since we wish to find the top
, we wish to find such that .
(2) Use lookup table to find
.
(3) Given that
, solve for .
(c)
(1) Interpret problem.
Since we wish to find the 25th percentile, we wish to find
such that .
(2) Use lookup table to find
.
(3) Given that
, solve for . Link to original
05
Normal distribution - cars passing toll booth
The number of cars passing a toll booth on Wednesdays has a normal distribution
. (a) What is the probability that on a randomly chosen Wednesday, more than 1,400 cars pass the toll booth?
(b) What is the probability that between 1,000 and 1,400 cars pass the toll booth on a random Wednesday?
(c) Suppose it is learned that at least 1200 cars passed the toll booth last Wednesday. What is the probability that at least 1300 cars passed the toll booth that day?
Link to originalSolution
14
(a)
(1) Write desired probability in terms of
values.
(2) Use lookup table to compute probability.
(b)
(1) Write desired probability in terms of
values.
(2) Use lookup table to compute probability.
(c)
(1) Set up conditional probability expression.
(2) Write desired probability in terms of
values.
(3) Use lookup table to compute probability.
Link to original