Problem 2

1. When Alice gets stressed out about her classes, she plays a game in which she rolls a pair of dice and if the sum of the two numbers is at least 10, she “wins.” For each win, she gives herself a dollar for ice cream.

(a) The night before the final exam, Alice will play this game 10 times. What is the probability that she will win herself at least $3 for ice cream?

(b) Alice teaches this game to Bob, who is taking another course. The night before his final exam, Bob plays the game 10 times. But Bob really likes ice cream, and he’s really stressed out, so he plays with a special pair of dice that are weighted so that p(5) = p(6) = 2p(4) = 2p(3) = 2p(2) = 2p(1). (In other words, he is twice as likely to roll a 5 or a 6 than any other number.) What is the probability that Bob will win himself at least $3 for ice cream?

(c) How many times should Alice play her game to ensure that she will have at least a 50% chance of winning at least $5? (NOTE: Alice will play with the original version in (a) not Bob’s cheating version in (b).) (HINT: This may be rather tedious to calculate by hand, so you may want to write a short program to calculate it. If you do, attach an image of your code as well as a short explanation of your procedure and the final answer.)

Game theory:

Analyze the second price" auction.
There are two bidders and one object being sold by auction. Each bidder
knows what the object is worth to him, but not what it is worth to the other
bidder. In other words, the object is worth v1 to bidder 1 and v2 to bidder 2.
Bidder 1 knows v1 but not v2, while bidder 2 knows v2 but not v1.
In the sealed bid second price auction, each bidder privately submits a bid
to the auctioneer. Let b1 be the bid submitted by bidder 1 and let b2 be the bid
submitted by bidder 2. The bidder submitting the highest bid wins the auction.
Rather than paying what he bid, the winning bidder pays the bid submitted by
the other bidder. Thus, if bi > bj , then bidder i's payo is vi ? bj and bidder
j's payo is zero. If they submit the same bid, then bidder 1 wins.

1. Can bidder i be worse o bidding bi > vi than bidding bi = vi?
2. Show that bidding higher than the bidder's valuation can never increase
his payoff. In other words, show that bidder i is never worse o bidding
bi = vi than bidding any bi > vi. ?
3. Can bidder j be worse o bidding bi < vi than bidding bi = vi?
4. Show that bidding lower than one's valuation can never increase a bidder's
valuation. In other words, show that bidder i is never worse o bidding
bi = vi than bidding any bi < vi. ?
5. What bid would you advise these bidders to submit?

Write a C program for the recursive algorithm that removes all occurrences of a specific character from a string. (please comment the code)

Java

Write a method that removes duplicates from an array of strings and returns a new array, free of any duplicate strings.

Two ordinary fair, six-sided dice ate rolled.
What is the probability the sum of the numbers on the two dice is 6, given that the number on at least one of the dice is 3?
What is the probability the sum of the numbers on the two dice is 8, given that the number on at least one of the dice is 3?
What is the probability that the sum of the numbers on the two dice is 9, given that it is not 3?
What is the probability that exactly one of the dice shows the number 1 given that the sum of the number is 3?

Suppose you roll a fair 15 sided die. The numbers 1-15 appear once on different sides. (Imagine a regular die with 12 sides instead of 6.) Answers may be left in formula form.

1. (a) What is the probability of rolling a 7?

2. (b) What is the probability of rolling an odd number and a number greater than 8?

3. (c) What is the probability of rolling an even number or a number greater than 9?

4. (d) Suppose you roll the die, record the number you see, and then roll it again. What is the probability that the sum of the die is 11?

5. (e) Suppose you roll the die, record the number you see, and then roll it again. What is the probability of rolling a 3 and then rolling a 3 again?

Suppose that Motorola uses the normal distribution to determine the probability of defects and the number of defects in a particular production process. Assume that the production process manufactures items with a mean weight of 10 ounces. Calculate the probability of a defect and the suspected number of defects for a 1,000-unit production run in the following situations.

1. To construct a particular binomial probability, it is necessary to know the total number of trials and the probability of success on each trial. TRUE OR FALSE

2. The mean of a binomial distribution can be computed in a "shortcut" fashion by multiplying n (the total number of trials) times π (the probability of success). TRUE OR FALSE

3. Judging from recent experience, 5% of the computer keyboards produced by an automatic, high-speed machine are defective. If six keyboards are randomly selected, what is the probability that none of the keyboards are defective?

4. On a very hot summer day, 5% of the production employees at Midland States Steel are absent from work. The production employees are randomly selected for a special in-depth study on absenteeism. What is the probability of randomly selecting 10 production employees on a hot summer day and finding that none of them are absent?

5. Which one of the following is NOT a condition of the binomial distribution?

Multiple Choice

A. Independent trials.

B. Only two outcomes.

C. The probability of success remains constant from trial to trial.

D. Sampling at least 10 trials.

The number of chromosomes in a diploid sexual species is 14. What is the exact probability that an individual of this species could produce by meiosis a gamete that includes all 7 chromosomes that the individual itself inherited from it's mother while excluding all 7 chromosomes inherited from its father?

A) Zero

B) 7/14 = 50%

C) 1/2 ⁷

D) 7 ^1/2

E) 1/7 (1 out of seven gametes) = 14%

F) There is no answer to this question

1. A number between 0 and 100,000 is chosen. What is the probability of choosing all the digits of the number greater than or equal to 4?