Sometimes I worry about finding something to write, but this week I’m torn by which a plethora of options. Last week I mentioned that I would talk a bit about the “Prosecutor’s Fallacy,” but since then the June issue of Scientific American arrived with a good article on non-rational optimum decisions. That sounds like an oxymoron, but is really an interesting aspect of decision making.

In the same issue Shermer takes Oprah to task for touting obvious nonsense. Either of these pieces could be the basis of commentary and might be the subject of future columns.

Then I also have on tap an interesting derivation of a simple result for potential gamblers. As a teaser, I’ll give you the answer and show the development later. The result is to give the number of events necessary to make even odds that something will happen. For instance, assume you want to know how many times to throw dice to have even chances of coming up snake eyes. For a long time, serious gamblers knew that 24 tosses had a slightly less than even chance of getting at least one instance of snake eyes, and 25 tosses had a slightly greater chance. This result was found experimentally be people with more time on their hands than mathematical background. You can predict the same thing much more easily by using this magic formula. Simply compute the odds of the event not happening on a single throw. In the case of snake eyes, that would be 35 to 1 since either die has 6 sides for 36 combinations, and only one of them is snake eyes (1, 1).

Multiply the odds by 0.7. That’s it. The result gives the number of throws to have even odds. In this case, 35 X 0. 7 = 24.5, which agrees with the experimental result. The magic 0.7 is a close approximation to the actual value, but it is good enough for general use.

Although I used dice probabilities for the example, this algorithm works for any event with a known probability.

How many cards would you have to draw to have even chances of drawing any particular card? When given this puzzle, most people say 26 since that is half a deck and the desired card is in one half or the other. If you draw 26 cards, you have half the deck, so there is a fifty-fifty chance that you have the desired card in your draw. Is that right? If not, why?

Now to the Prosecutor’s Fallacy, which is also called the Prosecutor’s Paradox: This is an example I used in my book. It comes from the OJ trial.

During the OJ trial, attorney Alan Dershowitz argued that OJ’s history of battering his wife was not relevant to the case based on the known statistics that out of 2,500 known batters, only 1 will be a murderer. [Note: the frequency of murders is assumed to be 45/100,000 based on known statistics.]

Is that argument justified? If you were Judge Ito making a decision about allowing the evidence to be presented based on this presentation, what would be your response?

Think about it before reading on, the results might surprise you.

Re-cast Mr. Dershowitz’s claim about wife-batterers and murderers in terms that most people understand more easily than quoting statistics. Assume a pool of 100,000 battered women. By the conditions presented above:

• 99,955 of them will not be murdered, and
• 45 will be murdered, of these
• 40 will be murdered by battering partners (100,000/2,500), leaving
• 5 to be murdered by others.

Therefore we conclude that 40/45 victims were killed by batterers. Yes, that means that 8 out of 9 murder victims were killed by someone who had previously battered them.

Does this analysis support Dershowitz’s conclusions about suppressing evidence? Think about the underlying assumption in this paradox/fallacy and wonder if you have ever been suckered in by a similar scam.

In response to the interest my original tutorial generated, I have completely rewritten and expanded it. Check out the tutorial availability through Lockergnome. The new version is over 100 pages long with chapters that alternate between discussion of the theoretical aspects and puzzles just for the fun of it. Puzzle lovers will be glad to know that I included an answers section that includes discussions as to why the answer is correct and how it was obtained. Most of the material has appeared in these columns, but some is new. Most of the discussions are expanded compared to what they were in the original column format.

Tags: probability, statistics, decision theory, algorithm, puzzle