Friday, September 26, 2008

Probability Paradoxes

(Updated September 29, 2014)

To close out our coverage of probability, let's look at three brain-teasers.

1. One is the famous "Birthday Paradox." Upon first learning that a group size of only 23 people is necessary for the probability to be .50 that two of the people will have the same birthday, most observers find this very counterintuitive. The Wikipedia's page on the topic may help clarify the key points. One of the approaches taken on the Wikipedia page uses the "n choose k" principle. Another approach elaborates on the "and/multiplication" principle. The probability of at least one pair of people having the same birthday is 1 minus the probability of no one having the same birthday. The latter can be thought of as the product of the following probabilities:

The first person definitely has his/her birthday on some day (1) times...

The second person having it on one of the other 364 days of the year (364/365) times...

The third person having his or hers on one of the remaining 363 days (363/365) times...

2. The second puzzle is the famous Monty Hall Problem (named after the host of the old game show "Let's Make a Deal"), which is described in detail here. To my mind, the clearest explanation of the surprising solution is that given by Leonard Mlodinow's book The Drunkard's Walk. Based on Mlodinow's writing, here's a diagram we created in class a few years ago (thanks to Kristina for taking the picture):


The basic idea is that scenarios in which switching helps occur twice as often as ones in which switching hurts. The Naked Statistics book has a mini-chapter devoted to the Monty Hall Problem. There are also various YouTube videos on the problem, such as this one.

3. Finally, the third brain-teaser involves winning the lottery twice. Statisticians emphasize the distinction between a particular, named individual winning twice (which had an estimated probability of 1-in-17 trillion in a New Jersey example) and the probability that someone, somewhere, sometime would win twice. The latter probability, because it takes into account the huge number of people who play the lottery and the frequency and volume of tickets sold, is estimated at something more like 1-in-30 (the exact calculations are not shown).

Based on the linked article, let's use the n-choose-k and multiplication/and rules to derive the 1-in-17 trillion probability for a particular individual.



Tuesday, September 23, 2008

Comparing the Olympic Swimming Times of Michael Phelps (2008) vs. Mark Spitz (1972) via z-Scores

I have now corroborated the results of our Michael Phelps/Mark Spitz Olympic swimming z-score class exercise, which I'll show below. This activity was inspired by an earlier study published in the Baseball Research Journal that used z-scores to compare home-run sluggers of different eras.

I shared the activity with two listserve discussion groups, those of the APA Division of Evaluation, Measurement, and Statistics and the Society for Personality and Social Psychology, offering to provide the raw data and documentation on how to conduct the exercise. I'm pleased to report that over 100 people have requested these materials to use in their own statistics classes. The materials can still be requested, via my faculty webpage (see link in the right-hand column). I framed the exercise as follows, in the documentation:

Michael Phelps, with eight gold medals in the 2008 Beijing Olympics (on top of six golds from the 2004 Athens games), and Mark Spitz, with seven gold medals in the 1972 Munich Olympics, are swimming’s two greatest champions.

The two swam many of the same events. Though the respective times by Phelps are several seconds faster than Spitz’s, the 36 years between 1972 and 2008 are a long time for improvements in training, technique, nutrition, and facilities. A statistic known as the z-score allows us to see which swimmer was more dominant relative to his contemporary peers.


Phelps and Spitz had three individual (non-relay) events in common, the 200-meter freestyle, 200-meter butterfly, and the 100-meter butterfly. Because I have a relatively small class and wanted to have groups of three or four students each work on a different segment of the data, we looked at only the first two of the aforementioned events. Two considerations to note are that (a) times were converted into total seconds to facilitate computations; and (b) where an athlete swam multiple races of the same event (i.e., heats, semifinals, and finals), his fastest time was used. Here are the results:

200 FREESTYLE

2008

Mean = 109.42 seconds
SD = 3.20
Phelps time = 102.96 seconds (1:42.96)
Phelps z = -2.02

[For non-statisticians who may be reading this, z = an individual's value minus the mean, with the difference then divided by the standard deviation. The latter represents how spread out the data are.]

1972

Mean = 120.33 seconds
SD = 4.57 seconds
Spitz time = 112.78 seconds (1:52.78)
Spitz z = -1.65

200 BUTTERFLY

2008

Mean = 117.85 seconds
SD = 3.01
Phelps time = 112.03 seconds (1:52.03)
Phelps z = -1.93

1972

Mean = 129.51 seconds
SD = 5.32
Spitz time = 120.70 seconds (2:00.70)
Spitz z = -1.66

Note that negatively signed z scores are a "good" thing, indicating by how much Phelps or Spitz was faster (i.e., consuming less time) than his respective competitors. As can be seen, Phelps was more dominating against the 2008 fields of his events, than Spitz was against the 1972 fields. It would also be interesting to look at the 100-meter freestyle, which of course, Phelps won by the narrowest of margins.

I thank Nancy Genero of Wellesley College, a fellow University of Michigan Ph.D., for sharing the results from her class; by comparing our respective data files for possible typographical errors, we were able to reconcile some minor differences. Also, as a technical note, an "outlier" swimmer who had a time of 2:33.75 in the 1972 200 freestyle (when the next slowest time was around 2:13) was excluded. An extreme value would have affected both the mean and SD, of course.

Unlike the above analyses, which used all competitors in an event (regardless of whether they reached the finals or even the semifinals), one could also look exclusively at the finals. To the extent that qualifying rules for the Olympics may have changed between 1972 and 2008, or that other factors were operative, the proportion of weak swimmers (in a world-class context) in the fields might have been different in the two Games, again possibly affecting the z-score results. University of Nevada Reno graduate student Irem Uz indeed analyzed only the finals, and these were his results:

Phelps 200 free z = -1.92
Spitz 200 free z = -1.34
Phelps 200 fly z = -1.62
Spitz 200 fly z = -2.01

Under this method, there's a little redemption for Spitz. Examining the results of Spitz's 200 fly win in 1972, his dominance is clear:

1. Mark Spitz 2:00.70 WR
2. Gary Hall 2:02.86
3. Robin Backhaus 2:03.23
4. Jorge Delgado, Jr. 2:04.60
5. Hans Faßnacht 2:04.69
6. András Hargitay 2:04.69
7. Hartmut Flöckner 2:05.34
8. Folkert Meeuw 2:05.57

The mean was roughly 2:04, putting Spitz 3.30 seconds faster than it. Meanwhile, the extremely tight clustering of the fourth- through eighth-place swimmers served to keep the overall SD small (1.63). The upshot is a very big z for Spitz.

ADDENDA

The Wall Street Journal's "Numbers Guy," Carl Bialik, provided some other types of Phelps-Spitz comparisons as this year's Olympics were going on.

The New York Times created an amazing slide show of graphics, showing how the swimming times of Phelps and Spitz stacked up against each other, and also how each fared against his respective competition.

Another blogger, Jeremy Yoder, independently came up with the idea to analyze z-scores for Phelps and Spitz. Yoder's results are different from the comparable analyses reported above, for some reason.

Here's a 2014 application of z-scores to golf.