Assignment 3 - Statistics. n n! (n r)!r! n = 1,2,3,...

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1 Assignment 3 - Statistics Name: Permutation: Combination: n n! P r = (n r)! n n! C r = (n r)!r! n = 1,2,3,... n = 1,2,3,... The Fundamental Counting Principle: If two indepndent events A and B can happen m and n different ways respective, then together they can happen in m n different ways. Problem: An access code has 6 characters. First four are digits and the last two are alphabets which are case sensitive. A thief trying to break this code has a probability of success: Problem: A student committee consists of 13 members. They need to elect a president, a vice president, and a treasurer. How many different ways this can be accomplished? Problem: You have the option of buying one car from 5 different types of cars and you may pick one insurance from 4 different choices. What are total number of ways you can have the car and insurance combination? 1

2 Problem: Permutation and combination: What are your chances of winning the Mega Millions Lottery? You pick 5 numbers from 1 to 56 without replacement and 1 number from 1 to 46. Problem: Among the 13 managers the company laid off 3 managers with the highest salaries. Is there enough justification to claim that the company laid off people to save money as opposed to randomly laying off 3 people. Problem: Answer the following questions: 1. How many different ways can 7 people sit on 7 different chairs? 2. How many different ways can 5 people sit on 7 different chairs? 3. How many different groups of 5 people are possible that can occupy 7 different chairs? 2

3 Expected Value E(x) = xp(x) Problem: If you toss a fair coin many many times, in theory infinitely many times, then what is your expected value? Assume that head is represented by 0 and tail by 1. Problem: If you roll a die what is the expected value? Problem: American Roulette is a form of gambling where there are 38 total slots labeled 1 through 36 and 0, and 00. There are many ways of playing this game, but here are 3 possibilities; calculate expected return for a gambler for each of these categories: 1. You bet that a particular number shows up and the return is 35 times your bet 2. You bet that an even number shows up and the return is just your bet 3. You bet on two consecutive numbers and you win if the either of these shows up and the return is 17 times your bet 4. The moral of the story: The house always wins in the long run. 3

4 Problem: Insurance is one of the fundamental driving forces of modern economy. Just to give you an idea, consider healcare industry. It is the largest industry in the U.S. and currently have yearly transactions of about $3 trillion; it is fundamentally driven by health insurance. As a healthcare insurance provider, you sell health insurance for a yearly premium. The probability that a randomly selected person who buys health insurance from you will get sick and charge $3,000 per year is 60%. If the yearly insurance premium is $10,000 then who do you think wins, the insurance provider or the insurance buyer? Problem: Assume that you are investing $10,000 in one bond. There are two types of bonds available. The first bond gives you a 7% return with a default rate of 3% and the second bond gives you a return of 9% with a default rate of 5%. Which one these bonds would you consider for investing your $10,000 assuming that you want to maximize your profit. 4

5 Binomial Distribution Conditions for Binomial Distribution, a discrete probability distribution: 1. Events/trials can be counted such as 0,1,2,3, Fixed number of events/trials 3. Probability is constant from trial to trial 4. Two fixed outcomes 5. Events/trials are independent The formula for calculating probabilities associated with random variable x P(x) = We normally denote q = 1 p n! (n x)!x! px (1 p) n x x = 0,1,2,3,... Problem: A student takes a nultiple choice test consisting of 10 questions and 4 choices for each question. 1. Show that the above situation satisfies all the criteria for the Binomial Probability Distribution. 2. Find the probability that a student gets exactly 5 questions right by pure guessing 3. Find the probability that a student gets less than 2 questions right by pure guessing 5

6 4. Find the probability that a student gets more than 8 questions right by pure guessing 5. If 70% is the passing grade then what is the probability that a student passes the test by pure guessing? Problem: A new drug named CURAIDS that is 60% effective in extending the average life of an AIDS patient by twenty years. Five randomly selected AIDS patients from Africa are treated with this new drug. Answer the following questions based on the above information: 1. Show that the above situation satisfies all the criteria for the Binomial Probability Distribution. 6

7 2. Find all the probabilities P(x) associated with the random variable x. Show your calculations. 3. What is the probability that no more than 4 patients are cured? Use results from part (b), do not do the calculations again. 4. Find the probability that more than 2 patients or less than or equal to 5 patients are cured. Use results from part (b), do not do the calculations again. 5. Find the probability that at least 4 patients are cured. Use results from part (b), do not do the calculations again. 6. Find probability that less than 2 or more than 3 patients are cured. Use results from part (b), do not do the calculations again. 7

8 Poisson Distribution Conditions for Poisson Distribution, a discrete probability distribution: 1. Countable events/trials such 0, 1, 2, 3, Events are independent meaning occurance of one event does not change the probability of another event 3. The mean or the expected value over an interval or unit, such as time, area, volume etc., is known or can be calculated from the past data 4. It is possible to count the number of events that have occurred, but does not make sense to ask how many did not occur (The fundamental difference between Binomial and Poisson distribution) Problem: Texting is part of modern life. A teenager communicated 10,000 text messages over the last year. Given this you are interested in finding probabilities associated with different numbers of text messages that a teenager can receive in a day. Show that this will conform to the conditions of Poisson Distribution and find the following: 1. Find the probability that the teenager will receive exactly 30 messages today 2. Find the probability that the teenager will receive between 28 and 30 messages today 8

9 Problem: In Dhaka, a city in Bangladesh, there were 4013 drug related crimes over one year period. Find the probability that on a given day there will be exactly 15 drug related crimes in that city. Use Poisson distribution. Explain the requirements for Poisson distribution. Problem: An emergency care facility at a hospital gets about 4000 patients over a period of one year. Find the probability that they will get 12 patients today. Problem: Banks serve more clients during the busy lunch hour. A bank services 45 customers on the average during the lunch hour. What is the probablity that tomoorow 50 customers will show up during the lunch hour. 9