I've been reading Warren Weaver's *Lady Luck*, an old classic in the field of probability and statistics. Since it was written for the "nontechnical" reader, and there aren't too many people with an amateur's interest in its subject, you probably haven't heard of the book. I'm here to say that it is informative, interesting, and, potentially, profitable. *Lady Luck* would be worth incalcuably more than its price at Amazon if it only persuaded you not to go to the casino or buy lottery tickets. But, as I have mentioned elsewhere (May 22 and May 28), the solutions to probabilistic problems are often profoundly counter-intuitive, which brings into view the possibility that the clever may invent betting games that friends and acquaintances will be eager to play despite the fact that they cannot win.

For example, you might find it rewarding to recruit people willing to play the following game:

You'll need a table filled with numeric information, such as is found all the time nowadays in Excel spreadsheets. The numbers should refer to things that can be counted--people, dollars, ballots, etc. (For reasons that will become obvious, the game would not be "fair" if played, say, with the batting averages of baseball players, or the values of the natural logarithmic function for the first hundred integers greater than 2.) Naturally the first digit of each number in the data set will be one of the first nine counting numbers: 1, 2, 3, 4, 5, 6, 7, 8 or 9. The game is to bet on which number turns up. There are nine possibilities, but to be "generous" you offer to play without odds and to pay on 5, 6, 7, 8 and 9--that is, on five of the nine possibilities, or on fractionally more than 55% of the presumably random results. Your opponent pays only on 1, 2, 3, and 4, and, to be yet more generous, you offer to let him select the data, only with the sensible stipulation that he may not inspect it beforehand.

Suppose he chooses the number of registered voters in Minnesota, by county, on November 1, 2006. If you click here and count, you should be persuaded that, despite your evident generosity, this game is very much in your favor: indeed if the bets were all one dollar you would be $31 to the good by the time the game had been played for all 87 Minnesota counties. Perhaps you suspect that I had to look high and low to find data tilted so much in favor of the player winning on 1, 2, 3 and 4. I assure you I merely selected a table that was near at hand, but there is no reason to take my word for it. Conduct your own experiments.

The fact is, as Weaver patiently explains, that the first digits of random numbers are not evenly distributed. It is hard to get your mind around why this should be so but, with Weaver's assistance, I'll try. Let us suppose you are filling a hat with all the counting numbers and then at some point leaving off to play the game with all the numbers that are so far "in play." After the first nine are in the hat, you would win 4 and lose 5. Notice, however, that you would win the next 40 in a row, so that when the hat has 49 items you would win almost 90% of the bets. Then you would lose the next 50 and be back down to 4 out of every 9. This pattern continues through all the powers of 10: the proportion of the numbers n and under starting with 1, 2, 3 or 4 oscillates between 4/9 (near 10, 100, 1000, etc.) and about 8/9 (near 49, 499, 4999, etc.). Since 0.4444 is very much closer to 0.5 than is 0.8888, you may begin to suspect that, notwithstanding appearances, the game we are considering is indeed very much in favor of the bettor who wins "only" on 1, 2, 3 and 4.

It turns out that the proportion of numbers with a first significant digit of n or less is not, as almost everyone would suppose, n/9, but is given rather by the base 10 logarithm of n + 1. So the precise probability of winning a single bet in our game is the log of 5, or 0.69897--essentially 70%. Since the log of 4 is equal to about 0.6, you could, as a fall back proposal when dealing with a potential foe who seems suspicious of your generosity, offer to pay on 4 in addition to 5, 6, 7, 8 and 9. Your winnings will steadily accrue but with not quite the same alacrity. Weaver fiendishly suggests that, when your foe is tired of losing, you could propose a switch to the phone book: he wins when the first nonzero digit after the dash is a 1, 2, 3 or 4, while big-hearted you now offers to win "only" on the comparatively rare occurrences of 5, 6, 7, 8 and 9.

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