What is a fair price to charge to break even

Assignment Help Applied Statistics

Reference no: EM132225704

Statistics Assignment - Discrete Random Variable

Question - Solve all questions.

Probability Distribution Function (PDF) for a Discrete Random Variable

Q1. Suppose that the PDF for the number of years it takes to earn a Bachelor of Science (B.S.) degree is given in table.

x	P(x)
3	0.05
4	0.40
5	0.30
6	0.15
7	0.10

a. In words, define the random variable X.

b. What does it mean that the values zero, one, and two are not included for x in the PDF?

Mean or Expected Value and Standard Deviation -

Q2. A game involves selecting a card from a regular 52-card deck and tossing a coin. The coin is a fair coin and is equally likely to land on heads or tails.

If the card is a face card and the coin lands on Heads, you win $6
If the card is a face card and the coin lands on Tails, you win $2
If the card is not a face card, you lose $2, no matter what the coin shows.

a. Find the expected value for this game (expected net gain or loss).

b. Explain what your calculations indicate about your long-term average profits and losses on this game.

c. Should you play this game to win money?

Q3. Complete the PDF and answer the questions.

x	P(x)	xP(x)
0	0.3
1	0.2
2
3	0.4

a. Find the probability that x = 2.

b. Find the expected value.

Q4. 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.

a. Construct a PDF for each investment.

b. Find the expected value for each investment.

c. Which is the safest investment? Why do you think so?

d. Which is the riskiest investment? Why do you think so?

e. Which investment has the highest expected return, on average?

Q5. Suppose that the PDF for the number of years it takes to earn a Bachelor of Science (B.S.) degree is given as in Table.

x	P(x)
3	0.05
4	0.40
5	0.30
6	0.15
7	0.10

On average, how many years do you expect it to take for an individual to earn a B.S.?

Q6. A "friend" offers you the following "deal." For a $10 fee, you may pick an envelope from a box containing 100 seemingly identical envelopes. However, each envelope contains a coupon for a free gift.

Ten of the coupons are for a free gift worth $6.
Eighty of the coupons are for a free gift worth $8.
Six of the coupons are for a free gift worth $12.
Four of the coupons are for a free gift worth $40.

Based upon the financial gain or loss over the long run, should you play the game?

a. Yes, I expect to come out ahead in money.

b. No, I expect to come out behind in money.

c. It doesn't matter. I expect to break even.

Q7. In a lottery, there are 250 prizes of $5, 50 prizes of $25, and ten prizes of $100. Assuming that 10,000 tickets are to be issued and sold, what is a fair price to charge to break even?

Binomial Distribution -

Q8. Recently, a nurse commented that when a patient calls the medical advice line claiming to have the flu, the chance that he or she truly has the flu (and not just a nasty cold) is only about 4%. Of the next 25 patients calling in claiming to have the flu, we are interested in how many actually have the flu.

Find the probability that at least four of the 25 patients actually have the flu.

Q9. People visiting video rental stores often rent more than one DVD at a time. The probability distribution for DVD rentals per customer at Video To Go is given Table. There is a five-video limit per customer at this store, so nobody ever rents more than five DVDs.

x	P(x)
0	0.03
1	0.50
2	0.24
3
4	0.07
5	0.04

a. Describe the random variable X in words.

b. Find the probability that a customer rents three DVDs.

c. Find the probability that a customer rents at least four DVDs.

d. Find the probability that a customer rents at most two DVDs.

Using the following information for next two questions -

The probability that the San Jose Sharks will win any given game is 0.3694 based on a 13 year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games.

Q10. The expected number of wins for that upcoming month is:

a. 1.67

b. 12

c. 382/1043

d. 4.43

Let X = the number of games won in that upcoming month.

Q11. What is the probability that the San Jose Sharks win at least five games in that upcoming month?

a. 0.3694

b. 0.5266

c. 0.4734

d. 0.2305

Q12. A student takes a 32 question multiple-choice exam, but did not study and randomly guesses each answer. Each question has three possible choices for the answer. Find the probability that the student guesses more than 75% of the questions correctly.

Q13. More than 96 percent of the very largest colleges and universities (more than 15,000 total enrollments) have some online offerings. Suppose you randomly pick 13 such institutions. We are interested in the number that offer distance learning courses.

a. In words, define the Random Variable X.

b. List the values that X may take on.

c. Give the distributions of X. X~_____(_____,_____)

d. On average, how many schools would you expect to offer such courses?

e. Find the probability that at most ten offer such courses.

f. Is it more likely that 12 or that 13 will offer such courses? Use numbers to justify your answer numerically and answer in a complete sentence.

Q14. At The Fencing Center, 60% of the fencers use the foil as their main weapon. We randomly survey 25 fencers at The Fencing Center. We are interested in the number of fencers who do not use the foil as their main weapon.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~_____(_____,_____).

d. How many are expected to not to use the foil as their main weapon?

e. Find the probability that six do not use the foil as their main weapon.

f. Based on numerical values, would you be surprised if all 25 did not use foil as their main weapon? Justify your answer numerically.

Q15. The chance of an IRS audit for a tax return with over $25,000 in income is about 2% per year. We are interested in the expected number of audits a person with that income has in a 20-year period. Assume each year is independent.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~_____(_____,_____)

d. How many audits are expected in a 20-year period?

e. Find the probability that a person is not audited at all.

f. Find the probability that a person is audited more than twice.

Q16. There are two similar games played for Chinese New Year and Vietnamese New Year. In the Chinese version, fair dice with numbers 1, 2, 3, 4, 5, and 6 are used, along with a board with those numbers. In the Vietnamese version, fair dice with pictures of a gourd, fish, rooster, crab, crayfish, and deer are used. The board has those six objects on it, also. We will play with bets being $1. The player places a bet on a number or object. The "house" rolls three dice. If none of the dice show the number or object that was bet, the house keeps the $1 bet. If one of the dice shows the number or object bet (and the other two do not show it), the player gets back his $1 bet, plus $1 profit. If two of the dice show the number or object bet (and the third die does not show it), the player gets back his $1 bet, plus $2 profit. If all three dice show the number or object bet, the player gets back his $1 bet, plus $3 profit. Let X = number of matches and Y = profit per game.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X~_____(_____,_____)

d. List the values that Y may take on. Then, construct one PDF table that includes both X & Y and their probabilities.

e. Calculate the average expected matches over the long run of playing this game for the player.

f. Calculate the average expected earnings over the long run of playing this game for the player.

g. Determine who has the advantage, the player or the house.

Q17. The literacy rate for a nation measures the proportion of people age 15 and over that can read and write. The literacy rate in Afghanistan is 28.1%. Suppose you choose 15 people in Afghanistan at random. Let X = the number of people who are literate.

a. Sketch a graph of the probability distribution of X.

b. Using the formulas, calculate the (i) mean and (ii) standard deviation of X.

c. Find the probability that more than five people in the sample are literate. Is it is more likely that three people or four people are literate.

Reference no: EM132225704

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