Due date: Tuesday 9/23, 11:59pm
Bernoulli process
01
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
02
06
Geometric distribution is memoryless
Suppose that
. Derive this equation:
Interpret the equation. (Inspired by the title.)
Link to originalSolution
06
(1) Set up conditional probability formula.
(2) Find formulas for numerator and denominator.
(3) Plug in values into initial formula
Link to original
03
07
Half the babies are female?
At Grace Community Hospital, 4 babies are delivered in one day. At Hope Valley Hospital, 6 babies are delivered in one day.
Consider these two events:
- (i) exactly half the babies born at Grace Community are female
- (ii) exactly half the babies born at Hope Valley are female
Perform a calculation to determine whether Event (i) is more probable, Event (ii) is more probable, or they are equally probable. (Assume the probability of each baby being born male or female is
.) Link to originalSolution
12
(i):
(ii):
So event (i) is more probable.
Link to original
Expectation and variance
04
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
Review problems
05
04
Rolling two dice
Two dice are rolled. Find the probabilities of the following events:
, the event that the sum is 10 , the event that the sum is 12 , the event that the two numbers are equal Link to originalSolution
01
(1) Consider the total number of outcomes.
Since there are two dice being rolled, there are 36 total outcomes.
(2) Consider the total number of desired outcomes for
. There are 3 total desired outcomes,
, , (6, 4).
(3) Use formula to find
.
(4) Consider the total number of desired outcomes for
. There is only 1 desired outcome,
.
(5) Use formula to find
.
(6) Consider the total number of desired outcomes for
. There are 6 total desired outcomes:
, , , , , Use formula to find
. Link to original
06
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
Optional challenge problems
- Complete any one of these problems instead of two of the above problems of your choice.
- You must indicate on your paper which of the above two should be replaced by this one. Indicate at both the above two problems (can otherwise leave blank) to direct the grader to the last page where you work your choice of challenge problem.
- Limit of one substitution.
07
05
A very strange car
A very strange car with
components will drive if at least half of its components work. Each component will work with the same probability , independently of the others. For what values of
is a car with more likely to drive than a car with ? (Start by defining a random variable that counts the number of working components.)
Link to originalSolution
05
(1) Define random variables.
Let
represent the car with three components. Let
represent the car with five components.
(2) Find probabilities that both cars work.
For the three-component car, we want
, so For the five-component car, we want
, so
(3) Find when
. Solve the inequality for
. Link to original
08
07
Binomial ratios
Suppose
.
- Find the value of
that maximizes . Do this by studying the successive ratios . - Use these ratios to compute
as a sum of 5 terms without using factorials. Do this by computing directly, and then writing a recursive algorithm that determines in terms of .
Link to originalSolution
07
(1) Find formula for ratio
.
(2) Interpret ratio.
We want
, so . Solving for
, we get . Since
is an integer, the is maximized when .
(3) Compute
directly. Based on the figure,
and .
(4) Use ratio to solve for successive terms.
(5) Add up probabilities.
Link to original
09
03
Expectation, variance of geometric variable
Derive formulas for
and given . Hint: For
you will get a sum that has terms like .
This series comes from the geometric series
(Differentiate both sides.)For
you will need to consider this general fact of algebra: (And apply the same methods as above.) Link to originalSolution
11
(1) State the PMF of a geometric random variable.
(2) Use formula for expectation to find
.
(3) Apply hint.
We have that
. Differentiating both sides yields
. Note that here,
, so
(4) Find expression
, and note that . Applying the hint, we have
. Using the linearity of expectation, we can write this as
.
(5) Find
and First, note that the second derivative of
is . Thus,
.
(6) Find
. Link to original