Probability and Statistics Unit Activity Answer Sheet Unit: Applying Probability The Lesson Activities will help you meet these educational goals: Mathematical Practices You will make sense of problems and solve them, reason abstractly and quantitatively, attend to precision, and look for and make use of structure Inquiry You will analyze data and communicate your results in written form STEM You will apply mathematical and technology tools and knowledge to analyze real-world situations 21st Century Skills You will employ online tools for research, use critical-thinking and problem-solving skills, and communicate effectively Introduction Take a look at this report produced by the Aviation Consumer Protection Division of the Office of Aviation Enforcement and Proceedings, which is a part of the US Department of Transportation Specifically, you re interested in table 2, on-time flight arrivals, which starts on page 6 and ends on page 9 The data in the table is from a typical month, so each percentage in the table can be thought of as a probability You can answer many probability questions using this table Directions and Analysis Task 1: Questions You Can Answer with Table 2 Refer to table 2 of the Air Travel Consumer Report to answer these questions a Given that a randomly chosen flight arrives in Atlanta (ATL), what is the probability that it arrives on time? The probability that a randomly chosen flight arrives in Atlanta (ATL) on time is 877% b Given that a randomly chosen US Airways (US) flight arrives in Denver (DEN), what is the probability that it arrives on time? The probability that a randomly chosen US Airways (US) flight arrives in Denver (DEN) on time is 861% c Given that a randomly chosen flight arrives in Los Angeles (LAX), what is the probability that the carrier is American Airlines (AA)? Copyright 2012 PLATO Learning, Inc All rights reserved 1
The probability that a randomly chosen flight arriving in Los Angeles (LAX) is an American Airlines (AA) flight is number of AA flights arriving in LAX 2,316 0137 137% number of flights arriving in LAX 16,924 d Chicago has two airports (MDW and ORD) If you re flying with Delta Airlines (DL) and your primary concern is to arrive on time, is it better to fly to MDW or to ORD? The probability that a DL flight arrives at MDW on time is 91% The probability that a DL flight arrives at ORD on time is 881% Since the probability of a DL flight arriving on time is greater at MDW, it is a better choice to fly to MDW e New York City has two airports (JFK and LGA) If you re flying with United Airlines (UA) and your primary concern is to arrive on time, is it better to fly to JFK or to LGA? The probability that a UA flight arrives at JFK on time is 922% The probability that a UA flight arrives at LGA on time is 846% Since the probability of a UA flight arriving on time is greater at JFK, it is a better choice to fly to JFK Task 2: Questions That Are Difficult or Nearly Impossible to Answer with Table 2 Many questions are difficult or nearly impossible to answer using table 2 in its current form Since computing the answers to these questions is difficult or close to impossible, just give the formulas you would need to obtain the answers to these questions a What is the probability that a randomly chosen flight arrives on time? The probability that a randomly chosen flight arrives on time is total flights arriving on time (all airports and all carriers) total flights arriving (all airports and all carriers) b Overall, which carrier has the highest probability of arriving on time? For each carrier, divide the carrier s total number of flights arriving on time at any airport by that carrier s total number of flights Then compare the probabilities of all the carriers to find which airline has the highest probability of arriving on time 2
c Given that a randomly chosen flight arrives on time, what is the probability that it arrives in Philadelphia (PHL)? The number of flights arriving on time at PHL is 884 6,113 5,404 100 The probability that a randomly chosen flight arrives at PHL, given that it arrives on time, number of flights arriving on time at PHL is total number of flights arriving on time d Given that the carrier is Hawaiian Airlines (HA), what is the probability that a randomly chosen flight arrives on time? The probability that a randomly chosen HA flight arrives on time is number of HA flights arriving on time total number of HA flight arrivals e If you re a frequent flyer who travels to many different airports, and you have the choice between SkyWest Airlines (OO) and Southwest Airlines (WN) for all your flights, which carrier should you choose if your primary concern is arriving on time? The probability that a randomly chosen OO flight arrives on time is number of OO flights arriving on time total number of OO flight arrivals The probability that a randomly chosen WN flight arrives on time is number of WN flights arriving on time total number of WN flight arrivals The airline carrier with the higher probability of arriving on time should be chosen 3
Task 3: Transforming Table 2 The reason it was difficult or nearly impossible to answer the questions in task 2 is that table 2 isn t quite a two-way table In other words, there is a total row at the bottom of the table, but there is no total column at the right of the table In this activity, you ll transform the table into a proper two-way table Open this spreadsheet The worksheet labeled Original contains the data as it appears in the report Scroll through the data in this worksheet and do a spot check to verify that it is, in fact, the same table This will give you some practice in navigating the table Once you are comfortable navigating, click the Modified tab In this worksheet, a few changes have been made to the table: Cells that contained H/ in the original table now contain zeros H/ indicated that a particular carrier didn t have a hub at a particular airport In other words, it had 0 arriving flights with 0% arriving on time For each airport, an additional column labeled # ON TIME has been added On the far right, you will find a TOTAL area with three new columns: # OF ARR, # ON TIME, and % ON TIME By filling in all of the blank columns, you ll transform the table into a two-way table Below you ll find instructions for filling in the blank columns to complete the two-way table Before you start, make a copy of the Modified worksheet in case you make a mistake One way to do this is to right-click the Modified tab, select Move or Copy on the shortcut menu, select the Create a copy checkbox, and click OK This procedure may vary slightly depending on your version of Microsoft Excel If you make a mistake, just delete the Modified tab you created by right-clicking it and selecting Delete from the shortcut menu Then make a new copy of the Modified worksheet and start again To complete the # ON TIME column for each airport: Click in cells C8 C22 and look at the formula bar to get an understanding of the formulas in these cells Do the same with cell C23 Copy cells C8 C23, and paste them into the space for cells F8 F23 Verify that all the formulas are doing what you expect Repeat this procedure to fill in the # ON TIME column for each individual airport To complete the TOTAL area on the right: Go to columns CK and CL at the far right of the table Examine what the formulas do for rows 8 22 Now examine the formula in cells CK23 and CL23 Click in Cell CM8 Enter a formula in the formula bar that will calculate what percentage of total arrivals are on-time arrivals Once you re satisfied this formula works, copy it into all the cells in that column (CM8 CM22, plus CM23) 4
Task 4: Questions That Can Now Be Answered by Table 2 Remember those questions that were difficult or nearly impossible to answer with table 2? See if you can answer them now, using the spreadsheet you modified a What is the probability that a randomly chosen flight arrives on time? The probability that a randomly chosen flight arrives on time is 863% b Overall, which carrier has the highest probability of arriving on time? The carrier with the highest probability of arriving on time is Virgin American (VX), which has a 916% probability of arriving on time c Given that a randomly chosen flight arrives on time, what is the probability that it arrives in Philadelphia (PHL)? The probability that a randomly chosen flight arrives at PHL, given that it arrives on time, is flights arriving on time at PHL 5,404 0021 21% total flights arriving on time 259,739 d Given that the carrier is Hawaiian Airlines (HA), what is the probability that a randomly chosen flight arrives on time? The probability that a randomly chosen flight arrives on time, given that the carrier is HA, is 72% e If you re a frequent flyer who travels to many different airports, and you have a choice between SkyWest Airlines (OO) and Southwest Airlines (WN) for all your flights, which carrier should you choose if your primary concern is arriving on time? The probability that a randomly chosen OO flight arrives on time is 842% The probability that a randomly chosen WN flight arrives on time is 888% I would choose Southwest Airlines because it has a higher probability of arriving on time 5
Task 5: Applying Conditional Probability to a Real-World Scenario Gary Foshee created a popular probability puzzle that goes like this: I have two children One is a boy born on a Tuesday What is the probability I have two boys? In this puzzle, knowing that one of Gary s children was born on a Tuesday is as important as knowing that he has one boy Assuming that having boys and girls are equally likely and that births are equally likely on every day of the week, what is the probability that Gary has two boys, given the available information? (Hint: When Gary said that one is a boy born on a Tuesday, he meant at least one child is a boy born on a Tuesday) There are four possibilities for pairs of children: boy-boy, girl-girl, boy-girl, and girl-boy There are 49 possibilities for pairs of days of the week Taking into account the given information, there are 13 possible pairs of days of the week for boy-boy: Monday-Tuesday, Tuesday-Tuesday, Wednesday-Tuesday, Thursday-Tuesday, Friday-Tuesday, Saturday-Tuesday, Sunday-Tuesday, Tuesday-Monday, Tuesday- Wednesday, Tuesday-Thursday, Tuesday-Friday, Tuesday-Saturday, and Tuesday-Sunday There are seven possibilities of boy-girl: Tuesday-Monday, Tuesday-Tuesday, Tuesday- Wednesday, Tuesday-Thursday, Tuesday-Friday, Tuesday-Saturday, and Tuesday-Sunday There are seven possibilities of girl-boy: Monday-Tuesday, Tuesday-Tuesday, Wednesday- Tuesday, Thursday-Tuesday, Friday-Tuesday, Saturday-Tuesday, and Sunday-Tuesday So, the total sample space consists of 27 possibilities, and there are 13 possible pairs of days 13 of the week for boy-boy Hence, the probability that Gary has two boys is 27 Resources Air Travel Consumer Report Hype about Conditional Probability Puzzles 6
Evaluation This project will be evaluated on a rubric that is based on the completeness, clarity, and thinking you exhibit in the Directions and Analysis section above Total Points: 10 Task 1: Questions You Can Answer with Table 2 Identify and derive the conditional probabilities Task 2: Questions That Are Difficult or Nearly Impossible to Answer with Table 2 Formulate solutions to conditional probability problems Task points: 2 2 points Task points: 2 2 points Task 3: Transforming Table 2 Transform the table into a two-way table Task 4: Questions That Can Now Be Answered by Table 2 Derive the conditional probabilities Task 5: Applying Conditional Probability to a Real-World Scenario Compute the boy-boy conditional probability Task points: 1 1 point Task points: 2 2 points Task points: 3 3 points 7