Would you like to have a lucky penny? Here’s how:
Start with a thousand pennies. Flip them all. Approximately 500 will come up heads. Discard the ones that came up tails, and flip the 500 heads again. Approximately 250 of them will come up heads. Discard the others.
Now flip the remaining 250 one more time. Approximately 125 will come up heads. Discard the tails and flip the heads again.
If you do this 10 times, on average there will be 1 penny left. It has the distinction of having come up heads 10 times in a row. So this is clearly a lucky penny. It always comes up heads.
Unfortunately, the argument is fallacious. This penny is not guaranteed to come up heads the next time it is flipped. The “lucky penny” is no more or less lucky than any other penny. The next time it is flipped, it might just as well come up tails as heads.
It just happens that the odds against a penny’s coming up heads 10 times in a row are about 1,000 to 1, which means that in a thousand chances, it is likely to happen once. The “lucky penny” just happens to be that once.
That much seems clear. What isn’t clear is that psychologically we tend to place a lot of emphasis on the one penny in a thousand that comes up heads 10 times in a row, and we ignore the 999 other pennies.
The lucky penny fallacy is all around us. Baseball statistics, for example.
Baseball fans and baseball announcers look for reasons to explain behavior that is really little more than statistical variation.
A .300 hitter in baseball on average gets 3 hits every 10 times up. But this does not mean that he will get exactly 3 hits every 10 times up. In fact, it would be extraordinary if he did.
There are random variations in each hitter’s output. These variations are mistakenly called hot streaks or slumps, which gives them a significance they do not deserve.
Harry Roberts, a statistician at the University of Chicago, has analyzed major league batting records and concluded that batters are no more likely to get a hit in his next at-bat during a hot streak as during a slump. Amos Tversky, a Stanford psychologist, has made a similar study of professional basketball players and found that they are no more likely to score a basket when they are “hot” than when they are “cold.” According to Tversky, the idea of the “hot hand” is a myth, just like the lucky penny. It fails to recognize statistical oddities for what they are.
This same analysis can be applied to the stock market, where, amid the mass of statistics generated each day, week, month and year, there are always a few statistical anomalies that seem to require explanation or investment. These are the “lucky pennies” of the stock market, the one-in-a-thousand chance that just happened to come in that day. Just random variation.
“Very often,” Tversky says, “the search for explanation in human affairs is a rejection of randomness.”
Perhaps you have heard of Crook County, Oregon. If you haven’t already, you will be hearing about it shortly. Crook County, in central Oregon, is the nation’s last bellwether county. Since it was created in 1882, it has voted for the winning candidate in each of the ensuing 26 presidential elections.
Should the country spare itself the time and expense of holding a national presidential campaign and election this year and just let the good people of Crook County elect our next president? After all, they always vote for the winner.
No more than the lucky penny is bound to come up heads next time it is flipped. The analogy is not perfect because which candidate a county votes for is not just a matter of chance, like flipping a coin. In each election it is reasonable to assume that more counties vote for the winner than for the loser.
It is worth noting that there are 3,106 counties in the United States. So it is not surprising--just as a matter of statistics--that one of them has voted for the winning presidential candidate for a century.
Note, too, that in 1984, there were two bellwether counties. But Palo Alto County in Iowa voted for Walter Mondale over Ronald Reagan that year. That now leaves us with one.
The notion of a lucky county has no more validity than a lucky penny. You cannot predict the future by postdicting the past.