Discrete Random Variables Homework

Solve these problems, then check your answers against the given solutions.

Exercises

Exercise 1
A venture capitalist, willing to invest $1,000,000, has three investments to choose from. The first investment, a software company, has a 10% chance of returning $5,000,000 profit, a 30% chance of returning $1,000,000 profit, and a 60% chance of losing the million dollars. The second company, a hardware company, has a 20% chance of returning $3,000,000 profit, a 40% chance of returning $1,000,000 profit, and a 40% chance of losing the million dollars. The third company, a biotech firm, has a 10% chance of returning $6,000,000 profit, a 70% of no profit or loss, and a 20% chance of losing the million dollars.

  1. Construct a PDF for each investment.
  2. Find the expected value for each investment.
  3. Which is the safest investment? Why do you think so?
  4. Which is the riskiest investment? Why do you think so?
  5. Which investment has the highest expected return, on average?

Exercise 2
Suppose that 20,000 married adults in the United States were randomly surveyed as to the number of children they have. The results are compiled and are used as theoretical probabilities. Let X = the number of children

X P(X = x)
 X · P (X = x)
0 0.10
1 0.20
2 0.30
3

4 0.10
5 0.05
6 (or more)
 0.05

Table 4.2

  1. Find the probability that a married adult has 3 children.
  2. In words, what does the expected value in this example represent?
  3. Find the expected value.
  4. Is it more likely that a married adult will have 2-3 children or 4 - 6 children? How do you know?

For each problem:
  1. In words, define the Random Variable X.
  2. List the values hat X may take on.
  3. Give the distribution of X. X~
Exercise 3
Six different colored dice are rolled. Of interest is the number of dice that show a "1". 
  1. On average, how many dice would you expect to show a "1"?
  2. Find the probability that all six dice show a "1".
  3. Is it more likely that 3 or that 4 dice will show a "1"? Use numbers to justify your answer numerically.

Exercise 4
A school newspaper reporter decides to randomly survey 12 students to see if they will attend Tet festivities this year. Based on past years, she knows that 18% of students attend Tet festivities. We are interested in the number of students who will attend the festivities.

  1. How many of the 12 students do we expect to attend the festivities?
  2. Find the probability that at most 4 students will attend.
  3. Find the probability that more than 2 students will attend.

Exercise 5
The chance of having an extra fortune in a fortune cookie is about 3%. Given a bag of 144 fortune cookies, we are interested in the number of cookies with an extra fortune. Two distributions may be used to solve this problem. Use one distribution to solve the problem.

  1. How many cookies do we expect to have an extra fortune?
  2. Find the probability that none of the cookies have an extra fortune.
  3. Find the probability that more than 3 have an extra fortune.
  4. As n increases, what happens involving the probabilities using the two distributions?

Exercise 6
A consumer looking to buy a used red Miata car will call dealerships until she finds a dealership that carries the car. She estimates the probability that any independent dealership will have the car will be 28%. We are interested in the number of dealerships she must call.

  1. On average, how many dealerships would we expect her to have to call until she finds one that has the car?
  2. Find the probability that she must call at most 4 dealerships.
  3. Find the probability that she must call 3 or 4 dealerships.

Exercise 7
On average, Pierre, an amateur chef, drops 3 pieces of egg shell into every 2 batters of cake he makes. Suppose that you buy one of his cakes.

  1. On average, how many pieces of egg shell do you expect to be in the cake?
  2. What is the probability that there will not be any pieces of egg shell in the cake?
  3. Let's say that you buy one of Pierre's cakes each week for 6 weeks. What is the probability that there will not be any egg shell in any of the cakes?
  4. Based upon the average given for Pierre, is it possible for there to be 7 pieces of shell in the cake? Why?

Exercise 8
You buy a lottery ticket to a lottery that costs $10 per ticket. There are only 100 tickets available be sold in this lottery. In this lottery there is one $500 prize, 2 $100 prizes and 4 $25 prizes. Find your expected gain or loss.


Exercise 9
A student takes a 10 question true-false quiz, but did not study and randomly guesses each answer. Find the probability that the student passes the quiz with a grade of at least 70% of the questions correct.


Exercise 10
A student takes a 32 question multiple choice exam, but did not study and randomly guesses each answer. Each question has 3 possible choices for the answer. Find the probability that the student guesses more than 75% of the questions correctly.

Source: Barbara Illowsky and Susan Dean, https://archive.org/details/CollaborativeStatisticsHomeworkBook/page/n67/mode/1up
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