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.