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Math 1F92 Assignment 4: Due Date Friday November 11 at 10:00 a.m. 2016 Instructions -All questions can be answered using 5.5, 6.1, 6.2 and 6.3

Math 1F92 Assignment 4: Due Date Friday November 11 at 10:00 a.m. 2016 Instructions -All questions can be answered using 5.5, 6.1, 6.2 and 6.3 in your textbook -Please be sure to show full work and steps. If any steps are missing, full marks cannot be rewarded. -Your assignment must be one-sided, stapled and with a cover page (which has your full name, box number, and student number). It must also be placed in an 9 x 12-inch envelope (which also has your full name, box number and student number) and unsealed. Failure to do any of these may result in a loss of marks. -The drop boxes for the assignment are located in Mackenzie Chown J Block on the fourth floor. Place your envelope in the drop box that contains your last name. Box 1- A-F Box 2- G-N Box 3- O-Z Questions 1) Certain automobile license plates consist of a sequence of three letters followed by three digits. a) How many possible license plates are there if no repetition of letters are permitted? b) How many possible license plates are there if no repetition of letters or digits are permitted? c) What is the probability of someone having a licence plate that starts with \"A\" and ends in \"3\" ? 2) From a group of 15 girls and 10 boys, a 7 person committee is to be formed. a) How many ways can it be formed with no restrictions? b) How many ways can a committee with 3 girls and 4 boys be formed? c) What is the probability of the committee consisting of Matilda and two other girls, as well as Derek and Hansel and two other boys? Math 1F92 Assignment 4: Due Date Friday November 11 at 10:00 a.m. 2016 3) Richard, a statistics major, doesn't like to go to the casino because he knows the expected winnings of any given game is negative. However, with his vast knowledge of how expectation and probability works, he created a game to play with his friend Jason (who has no idea what expectation or probability is). The game is as follows: Richard and a friend each roll a die. i) ii) iii) If the sum of the two dice are 7 or greater, Jason owes Richard $5. If the sum of the two dice are 5 or less, Richard owes Jason $10. If the sum of the two dice are 6, no amount of money is owed. a) Find the probability distribution (or model) of winnings for both Richard and Jason. b) Find the expected winnings for both Richard and Jason. Why isn't this game fair? c) If they played 180 games straight, what are the expected winnings for Richard and for Jason? d) Change the amount Richard should owe so the game becomes fair. HINT: What has to happen for it to be a fair game? 4) The game 'Sic Bo' is a Chinese game at the casino where 3 dice are rolled and then the sum of 3 dice are calculated. Most people who play this game like to bet on 'High' or 'Low' where the outcomes are as follows: High consists of the sum being {11,13,14,16,17} with a probability of 48.6% Low consists of the sum being {4,5,7,8,10} with a probability of 48.6% Triples consists of the sum being {3,6,9,12,15,18} with a probability of 2.8% a) Find the expected winnings and standard deviation if a $1 bet is placed on 'High' for this game. Compare this expectation with the expectation of winning when a $1 is placed on 'Red' in 'Roulette'. (Use the notes for the expectation for Roulette). b) Find the expected winnings for betting on 'High' in 'Sic Bo' as well as betting on 'red' in 'Roulette' for: i. ii. iii. iv. 50 games 100 games 500 games 1000 games? Math 1F92 Assignment 4: Due Date Friday November 11 at 10:00 a.m. 2016 c) Complete the following using excel and XLSTAT: i. Open the excel file 'Sic Bo and Roulette.xlsx'. ii. Under the tab 'Sic Bo' (should be default first), type this code in cell E2: '=LOOKUP(RAND(),$C$3:$C$4,$A$3:$A$4)'. Drag the value down to cell E51 (for 50 values). iii. Do the same for the F, G and H column dragging down to cell F101, G501 and H1001 respectively. You should now have 4 columns with either '-1' or '1' values (50 for the first, 100 for the second, etc.). iv. Create a histogram for each number of games for Sic Bo. Ensure the 'discrete' option is selected and choose '5' for Number of Intervals. Copy and paste each of them onto a Microsoft word Document so they fit on one page. v. Click the tab at the bottom entitled 'Roulette' and do steps ii-iv again with these probabilities. vi. You should now have 2 Microsoft Word documents (or one 2 paged document) with 4 histograms on each page (you may want to change the layout settings to landscape). Attach these pages at the back of your assignment. d) Using the histograms from part c) find the net winnings for 50, 100, 500 and 1000 games of both Roulette and Sic Bo. How did the values of each net winning compare with the expected winnings in part b) as the number of games increased? Net Winnings= \t[1 1 ] e) In the long run, do you think you'll lose less betting on 'High' for 'Sic Bo' or 'Red' for 'Roulette'? Explain. Did this happen with your empirical data? Math 1F92 Assignment 4: Due Date Friday November 11 at 10:00 a.m. 2016 5) Based on past history, an airline determines that an average of 5% of people with reservations don't show up for their flights. Consequently, the airline routinely overbooks flights, accepting 350 reservations for a plane that seats 250 passengers. This system can be modelled by a binomial distribution with n=350, and p = the probability that someone with a reservation does show up. a) What probability would you need to find to determine that, when 350 reservations are accepted for a particular flight, there are more passengers than seats? Use XLSTAT, the file 'airplane.xlsx' and the steps below to find this probability. Step 1) Under the 'P(X=x)' column in D1, type in '=binom.dist'. At this point, it should say binom.dist(number_s, trials, probability, cumulative). This is essentially saying the function needs 4 values separated by commas. Step 2) For number_s, click the '0' cell in C2. Step 3) For trials, click the '350' value in cell A2. Then add a \"$\" before the \"A\" and the \"2\". It should look like this: \"$A$2\". These dollar signs are going to keep the 'n' value the same for when we drag this formula. Step 4) For probability, click the '0.05' in cell B2. Then add a \"$\" before the \"B\" and the \"2\". It should now look like this: \"$B$2\". These dollar signs are going to keep the p value the same for when we drag the formula down. Step 5) For cumulative, type in \"FALSE\". It may even automatically come up when you start typing. Close the bracket and press \"enter\". You should get a formula Step 6) Click the cell. At the bottom right hand corner of the cell there should be a green square. Drag it all the way down to the bottom to get all the probabilities. Step 7) Once you have all the probabilities. You can sum up the ones you are concerned with by using the \"=sum\" function, and selecting the particular cells. Write the final probability value down and explain how you knew which values to select. You do not need to print out any XLSTAT for this question. b) Based on your answer to a), is there need for concern, i.e. should the airline's policy on overbooking be changed? Explain. Math 1F92 Assignment 4: Due Date Friday November 11 at 10:00 a.m. 2016 6) In the 1990s, possible abuse of minorities by Philadelphia police officers was investigated. One study analyzed whether African-American drivers were more likely than others to be targeted for traffic stops. They studied the results of 262 independent police car stops during one week in 1997 and found that 207 of the drivers stopped were AfricanAmerican. At that time, 42.2% of Philadelphia's population was African-American. a) Find the mean and standard deviation for the random variable X, the number of African-Americans stopped in 262 traffic stops. b) Interpret the mean. c) Is there evidence to suggest that there was racial profiling in Philadelphia at that time? Explain. 7) The potholes on a major highway in the city of Chicago occur at the rate of 3.4 per mile. Compute the probability that the number of potholes over 3 miles of a randomly selected highway is: a) exactly seven b) fewer than seven c) at least seven d) Would it be unusual for a randomly selected 3-mile stretch of highway in Chicago to contain more than 15 potholes? Explain

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