Saturday, August 25, 2018

"What are the odds?" You Have the Same Taxi Driver Twice

Let's say you're traveling to a major city, and decide to use taxis the entire time you're there (declining both public transit and Uber/Lyft). At minimum, there are thousands of taxi drivers in all major cities in the US, as well as around the world. So what are the odds that you encounter the same taxi driver twice during your trip?

For our first example, let's use New York City. Let's assume that you're spending a full week (7 days) in the city, and you take 4 taxi trips per day, or 28 trips in total. There are 13,587 medallions in existence, which results in a very unlikely scenario you'll have the same driver twice (assuming that the chances of seeing any given driver is equal, which isn't reflective of reality since drivers aren't evenly distributed throughout a city, but we'll roll with this assumption for ease of calculation).

This actually is analogous to the birthday paradox. In a room of 23 people, there's a 50/50 chance of two people having the same birthday. That's because there are 253 different possible combinations of birthdays within those 23 people, and the chance of 2 people NOT sharing a birthday is 364/365. But multiply that out 253 times and you get (364/365)^253, or 49.95%. The chances that at least one of those pairs match is 1 - 49.95%, or 50.05%.

We can use a similar formula to determine the likelihood that you end up with the same taxi driver again:

n is the number of trips (think of them as pairs of trips) and in this case the probability of NOT finding a match would be (# of drivers - 1) / # of drivers. In the New York example, that's 13586/13587. Over 28 rides, you get: 1 - (13586/13587)^((28(28-1))/2) = 2.74%

How many rides would you need to take to get the chances to 50/50? The square root of n is approximately the number of rides you would need to take to get even odds of a match, although it undershoots it a little bit. For NY, you would need 138 rides to get to there, or 35 days at 4 rides/day.

 City Country # Taxi Drivers # Days Trips/Day Total Trips Prob Same Driver # of Trips to Expect Repeat San Francisco USA 1,825 7 4 28 18.71% 51 Houston USA 2,245 7 4 28 15.50% 56 Los Angeles USA 2,300 7 4 28 15.16% 57 Washington DC USA 6,300 7 4 28 5.82% 94 Chicago USA 6,650 7 4 28 5.53% 97 New York City USA 13,587 7 4 28 2.74% 138

As with my previous post about a doctor on an airplane, the chances entirely depend upon the number of drivers in the city. The less drivers, the more likely you see one twice.

What about around the world? This thought problem goes to the absolute extreme in Mexico City, which has one of the largest taxi fleets in the world with 140,000 taxis!

 City Country # Taxi Drivers # Days Trips/Day Total Trips Prob Same Driver # of Trips to Expect Repeat Toronto Canada 4,849 7 4 28 7.50% 83 Tokyo Japan 35,000 7 4 28 1.07% 222 Bogota Colombia 53,000 7 4 28 0.71% 273 Beijing China 68,500 7 4 28 0.55% 310 London England 138,957 7 4 28 0.27% 442 Mexico City Mexico 140,000 7 4 28 0.27% 443

Thursday, August 23, 2018

Preseason NCAAF Rankings for 2018

As I did last yearthe year beforethe year before, and the year before that, I've created a new set of preseason NCAAF rankings that take into account player turnover and recruiting classes. The following is literally copied and pasted from last year's write-up so you don't have to click that first link:

As before, I used my final Composite ratings from the MDS Model (from last season) as the base (which takes into account both a forward-looking predictive component and a past-performance only retrodictive component), and then factored in ESPN's Preseason FPI and the S&P+ projections, both of which take into account player changes on each team. Once the season starts, this "preseason" rating will be faded out as the season progresses, carrying less and less weight with each ensuing week.

In the below list, the "Trend" indicates whether the respective team's new ranking rose or fell relative to last year's preseason ratings. We gain Liberty (moving up to FBS) and lose Idaho (moving down to FCS), keeping the number of teams at 130.

What a surprise! Alabama is #1!

 Rank Team PRESEASON Trend 1 Alabama 0.910 DOWN 2 Ohio State 0.885 UP 3 Clemson 0.883 UP 4 Washington 0.826 UP 5 Oklahoma 0.823 DOWN 6 Georgia 0.805 UP 7 Notre Dame 0.803 UP 8 Auburn 0.796 UP 9 Michigan 0.779 UP 10 Penn State 0.777 UP 11 Wisconsin 0.758 UP 12 Florida State 0.748 DOWN 13 Michigan State 0.739 UP 14 USC 0.739 DOWN 15 Stanford 0.735 DOWN 16 Miami (FL) 0.729 UP 17 LSU 0.729 DOWN 18 Mississippi State 0.709 UP 19 Oklahoma State 0.692 DOWN 20 Florida 0.682 DOWN 21 Oregon 0.680 DOWN 22 Texas A&M 0.679 UP 23 TCU 0.678 DOWN 24 Texas 0.667 UP 25 Ole Miss 0.658 DOWN 26 Louisville 0.656 DOWN 27 Virginia Tech 0.654 UP 28 Iowa 0.635 UP 29 Utah 0.626 UP 30 Boise State 0.623 UP 31 UCLA 0.623 DOWN 32 North Carolina State 0.614 DOWN 33 Northwestern 0.611 UP 34 Baylor 0.608 DOWN 35 Missouri 0.603 UP 36 South Carolina 0.599 UP 37 North Carolina 0.598 DOWN 38 Arkansas 0.588 DOWN 39 Washington State 0.587 DOWN 40 West Virginia 0.586 DOWN 41 Arizona 0.582 UP 42 Tennessee 0.582 DOWN 43 Georgia Tech 0.582 UP 44 Duke 0.581 UP 45 Pittsburgh 0.573 DOWN 46 Memphis 0.573 DOWN 47 California 0.572 UP 48 Texas Tech 0.572 UP 49 Nebraska 0.557 DOWN 50 Wake Forest 0.555 UP 51 Kansas State 0.553 DOWN 52 Iowa State 0.547 UP 53 Arizona State 0.547 DOWN 54 Boston College 0.544 UP 55 San Diego State 0.541 DOWN 56 Houston 0.537 DOWN 57 UCF 0.537 UP 58 South Florida 0.535 DOWN 59 Toledo 0.519 DOWN 60 Indiana 0.518 DOWN 61 Minnesota 0.517 DOWN 62 Kentucky 0.516 UP 63 Syracuse 0.508 DOWN 64 Brigham Young 0.501 DOWN 65 Purdue 0.501 UP 66 Appalachian State 0.491 DOWN 67 Marshall 0.491 UP 68 Florida Atlantic 0.481 UP 69 Maryland 0.477 UP 70 Vanderbilt 0.471 DOWN 71 Temple 0.471 DOWN 72 Navy 0.470 DOWN 73 Virginia 0.463 UP 74 Utah State 0.461 UP 75 Northern Illinois 0.460 UP 76 Louisiana Tech 0.460 UP 77 Western Michigan 0.460 DOWN 78 Fresno State 0.456 UP 79 Arkansas State 0.451 UP 80 Colorado 0.449 DOWN 81 Ohio 0.440 UP 82 Cincinnati 0.432 DOWN 83 Rutgers 0.432 UP 84 Middle Tennessee 0.425 UP 85 Southern Methodist 0.418 UP 86 Western Kentucky 0.417 DOWN 87 Troy 0.415 UP 88 Wyoming 0.412 UP 89 Illinois 0.402 DOWN 90 Tulsa 0.402 DOWN 91 Miami (OH) 0.397 UP 92 Colorado State 0.396 DOWN 93 Southern Miss 0.392 DOWN 94 Bowling Green 0.383 DOWN 95 Oregon State 0.377 DOWN 96 Georgia Southern 0.371 DOWN 97 Tulane 0.366 UP 98 Air Force 0.365 DOWN 99 Nevada 0.362 UP 100 North Texas 0.357 UP 101 Army 0.351 UP 102 Buffalo 0.351 UP 103 Kansas 0.347 UP 104 Central Michigan 0.344 DOWN 105 Massachusetts 0.341 UP 106 UNLV 0.329 UP 107 Eastern Michigan 0.328 UP 108 New Mexico 0.317 DOWN 109 UTSA 0.315 UP 110 New Mexico State 0.305 UP 111 Old Dominion 0.303 DOWN 112 Louisiana-Monroe 0.303 UP 113 East Carolina 0.303 DOWN 114 Akron 0.300 DOWN 115 UAB 0.296 UP 116 Florida International 0.283 DOWN 117 Georgia State 0.281 DOWN 118 Ball State 0.280 DOWN 119 South Alabama 0.279 DOWN 120 Connecticut 0.272 DOWN 121 Liberty 0.258 UP 122 San Jose State 0.252 DOWN 123 Louisiana-Lafayette 0.244 DOWN 124 Hawaii 0.228 DOWN 125 Coastal Carolina 0.227 UP 126 Charlotte 0.216 UP 127 Rice 0.213 DOWN 128 Kent State 0.212 DOWN 129 Texas State 0.211 UP 130 UTEP 0.168 DOWN