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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.


CityCountry# Taxi Drivers# DaysTrips/DayTotal TripsProb Same Driver# of Trips to Expect Repeat
San FranciscoUSA1,825742818.71%51
HoustonUSA2,245742815.50%56
Los AngelesUSA2,300742815.16%57
Washington DCUSA6,30074285.82%94
ChicagoUSA6,65074285.53%97
New York CityUSA13,58774282.74%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!


CityCountry# Taxi Drivers# DaysTrips/DayTotal TripsProb Same Driver# of Trips to Expect Repeat
TorontoCanada4,84974287.50%83
TokyoJapan35,00074281.07%222
BogotaColombia53,00074280.71%273
BeijingChina68,50074280.55%310
LondonEngland138,95774280.27%442
Mexico CityMexico140,00074280.27%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!


RankTeamPRESEASONTrend
1Alabama0.910DOWN
2Ohio State0.885UP
3Clemson0.883UP
4Washington0.826UP
5Oklahoma0.823DOWN
6Georgia0.805UP
7Notre Dame0.803UP
8Auburn0.796UP
9Michigan0.779UP
10Penn State0.777UP
11Wisconsin0.758UP
12Florida State0.748DOWN
13Michigan State0.739UP
14USC0.739DOWN
15Stanford0.735DOWN
16Miami (FL)0.729UP
17LSU0.729DOWN
18Mississippi State0.709UP
19Oklahoma State0.692DOWN
20Florida0.682DOWN
21Oregon0.680DOWN
22Texas A&M0.679UP
23TCU0.678DOWN
24Texas0.667UP
25Ole Miss0.658DOWN
26Louisville0.656DOWN
27Virginia Tech0.654UP
28Iowa0.635UP
29Utah0.626UP
30Boise State0.623UP
31UCLA0.623DOWN
32North Carolina State0.614DOWN
33Northwestern0.611UP
34Baylor0.608DOWN
35Missouri0.603UP
36South Carolina0.599UP
37North Carolina0.598DOWN
38Arkansas0.588DOWN
39Washington State0.587DOWN
40West Virginia0.586DOWN
41Arizona0.582UP
42Tennessee0.582DOWN
43Georgia Tech0.582UP
44Duke0.581UP
45Pittsburgh0.573DOWN
46Memphis0.573DOWN
47California0.572UP
48Texas Tech0.572UP
49Nebraska0.557DOWN
50Wake Forest0.555UP
51Kansas State0.553DOWN
52Iowa State0.547UP
53Arizona State0.547DOWN
54Boston College0.544UP
55San Diego State0.541DOWN
56Houston0.537DOWN
57UCF0.537UP
58South Florida0.535DOWN
59Toledo0.519DOWN
60Indiana0.518DOWN
61Minnesota0.517DOWN
62Kentucky0.516UP
63Syracuse0.508DOWN
64Brigham Young0.501DOWN
65Purdue0.501UP
66Appalachian State0.491DOWN
67Marshall0.491UP
68Florida Atlantic0.481UP
69Maryland0.477UP
70Vanderbilt0.471DOWN
71Temple0.471DOWN
72Navy0.470DOWN
73Virginia0.463UP
74Utah State0.461UP
75Northern Illinois0.460UP
76Louisiana Tech0.460UP
77Western Michigan0.460DOWN
78Fresno State0.456UP
79Arkansas State0.451UP
80Colorado0.449DOWN
81Ohio0.440UP
82Cincinnati0.432DOWN
83Rutgers0.432UP
84Middle Tennessee0.425UP
85Southern Methodist0.418UP
86Western Kentucky0.417DOWN
87Troy0.415UP
88Wyoming0.412UP
89Illinois0.402DOWN
90Tulsa0.402DOWN
91Miami (OH)0.397UP
92Colorado State0.396DOWN
93Southern Miss0.392DOWN
94Bowling Green0.383DOWN
95Oregon State0.377DOWN
96Georgia Southern0.371DOWN
97Tulane0.366UP
98Air Force0.365DOWN
99Nevada0.362UP
100North Texas0.357UP
101Army0.351UP
102Buffalo0.351UP
103Kansas0.347UP
104Central Michigan0.344DOWN
105Massachusetts0.341UP
106UNLV0.329UP
107Eastern Michigan0.328UP
108New Mexico0.317DOWN
109UTSA0.315UP
110New Mexico State0.305UP
111Old Dominion0.303DOWN
112Louisiana-Monroe0.303UP
113East Carolina0.303DOWN
114Akron0.300DOWN
115UAB0.296UP
116Florida International0.283DOWN
117Georgia State0.281DOWN
118Ball State0.280DOWN
119South Alabama0.279DOWN
120Connecticut0.272DOWN
121Liberty0.258UP
122San Jose State0.252DOWN
123Louisiana-Lafayette0.244DOWN
124Hawaii0.228DOWN
125Coastal Carolina0.227UP
126Charlotte0.216UP
127Rice0.213DOWN
128Kent State0.212DOWN
129Texas State0.211UP
130UTEP0.168DOWN