Assignment is attached in Word.
Stats Assignment 2 Probability Please show work, and/or provide explanations. 1. Given events A and B, P(A) = .6, P(B) = .4, and P(AB) = .76. a. Find P(AB). b. Find P(A|B). c. Are A and B independent events? Explain (or show why or why not). 2. Students are known to have an average height of 68 inches with a standard deviation of 3 inches. Assuming this population is distributed normally, what is the probability that a randomly selected student is: a. Between 71 and 74 inches tall? b. Over 76 inches tall? c. Exactly 68 inches tall? 3. The demographics of your company’s work force, shown by gender and hair color, are given in the following table. Black/Brown Blond Other Male 15 5 10 Female 5 10 5 a. What is the probability that a randomly selected employee is a blond female? b. What is the probability that a randomly selected employee is blond or a female? c. Given that a randomly selected employee is a female, what is the probability that she is blond? 4. Your company has 1001 employees, and you suspect you have one corporate spy among your employees. This spy has been bringing company documents home to sell to your competitors. However, from an employee survey you know that 5% of your (honest) workforce brings company documents home with them so they can work from home in the evenings. During an unannounced search of your employees as they leave work, if you discover an employee taking documents home, what is the probability that individual is the spy? 5. You purchase two external storage devices (Device 1 and Device 2) to back up your statistics project data. Each device has a 1% chance of failure. a. Identify the sample space (in terms of Failure or Non-failure) for this situation and define a random variable that reflects the number of copies of your data that you have available. b. Find the probability function for this random variable. c. Find the cumulative probability function for this random variable. d. What is the expected value of this random variable? e. What is the standard deviation of this random variable? f. What is the overall reliability of your data backup system? 6. Your company is hiring customer service representatives for a new call center. You are expecting to receive an average of 24 calls during any given hour. a. Assuming independence, how many calls do you expect in any five-minute period? b. What is the probability of receiving three calls in a five-minute period? c. What is the probability of receiving more than five calls in a five-minute period? d. How much time, in minutes, do you expect to wait between calls? e. What is the probability of two calls occurring within two minutes of each other? f. What is the probability of waiting more than three minutes between calls? 7. A cubic foot of soil contains on average one grain of thorium. What is the probability that a cubic yard of soil contains between 25 and 30 grains of thorium (inclusive)? Hint: 1 yd = 3 ft. 8. Twenty-five percent of Americans are college graduates. You survey a random sample of fifteen Americans. What is the probability that: a. Exactly three of them are college graduates? b. Between two and five (inclusive) of them are college graduates? c. More than seven of them are college graduates? 9. A standard deck contains 52 cards, four of which are aces. If you are randomly dealt a five-card hand, what is the probability of having exactly three aces? 10. You are working for a company that does third-party verification of securitized mortgages. You sample a number of loans from a bundle and you perform a complete due-diligence on every loan in the sample to determine if it gets a rating of “failure.” You receive a bundle of 100 loans that is claimed to have a “C” rating (no more than 10% will fail). You randomly sample ten of these loans to perform your complete review. a. How many of these loans would you expect to fail? b. What is the probability of 5 loans from your sample getting a rating of “failure?” c. If 5 out of your sample of 10 have a rating of “failure,” what are your next steps as the third-party verification source? 11. There are 52 cards in a standard deck. People have been shuffling card decks for centuries. This question involves the number of possible of shuffles (different arrangements of the cards as they lie in the deck). Please answer part “a” first. There is no credit associated with this part, but it is there to test your intuition. a. It is likely that all arrangements of card decks have been realized throughout the centuries that people have been shuffling cards. (T or F) b. If every person on earth were given a deck of cards, and each one shuffled their deck every second, and if every shuffle resulted in a DIFFERENT arrangement of the decks, how many years would it take for every possible arrangement of a deck of cards to be realized? (Assume a global population of 7 billion, and that a year consists of 365.25 days.)