statistical swindles
This section presents some examples of how one can be misled by arguments that
require one to ignore the calculus of probability.
Misleading Use of Statistics
The field of statistics has a poor image in the minds of many people because there is
a widespread belief that statistical data and statistical analyses can easily be manipulated in an unscientific and unethical fashion in an effort to show that a particular
conclusion or point of view is correct. We all have heard the sayings that “There
are three kinds of lies: lies, damned lies, and statistics” (Mark Twain [1924, p. 246]
says that this line has been attributed to Benjamin Disraeli) and that “you can prove
anything with statistics.”
One benefit of studying probability and statistics is that the knowledge we gain
enables us to analyze statistical arguments that we read in newspapers, magazines,
or elsewhere. We can then evaluate these arguments on their merits, rather than
accepting them blindly. In this section, we shall describe three schemes that have been
used to induce consumers to send money to the operators of the schemes in exchange
for certain types of information. The first two schemes are not strictly statistical in
nature, but they are strongly based on undertones of probability.
Perfect Forecasts
Suppose that one Monday morning you receive in the mail a letter from a firm
with which you are not familiar, stating that the firm sells forecasts about the stock
market for very high fees. To indicate the firm’s ability in forecasting, it predicts that a
particular stock, or a particular portfolio of stocks, will rise in value during the coming
week. You do not respond to this letter, but you do watch the stock market during the
week and notice that the prediction was correct. On the following Monday morning
you receive another letter from the same firm containing another prediction, this one
specifying that a particular stock will drop in value during the coming week. Again
the prediction proves to be correct.
52 Chapter 1 Introduction to Probability
This routine continues for seven weeks. Every Monday morning you receive a
prediction in the mail from the firm, and each of these seven predictions proves to
be correct. On the eighth Monday morning, you receive another letter from the firm.
This letter states that for a large fee the firm will provide another prediction, on
the basis of which you can presumably make a large amount of money on the stock
market. How should you respond to this letter?
Since the firm has made seven successive correct predictions, it would seem that
it must have some special information about the stock market and is not simply
guessing. After all, the probability of correctly guessing the outcomes of seven
successive tosses of a fair coin is only (1/2)7 = 0.008. Hence, if the firm had only been
guessing each week, then the firm had a probability less than 0.01 of being correct
seven weeks in a row.
The fallacy here is that you may have seen only a relatively small number of the
forecasts that the firm made during the seven-week period. Suppose, for example,
that the firm started the entire process with a list of 27 = 128 potential clients. On
the first Monday, the firm could send the forecast that a particular stock will rise in
value to half of these clients and send the forecast that the same stock will drop in
value to the other half. On the second Monday, the firm could continue writing to
those 64 clients for whom the first forecast proved to be correct. It could again send
a new forecast to half of those 64 clients and the opposite forecast to the other half.
At the end of seven weeks, the firm (which usually consists of only one person and a
computer) must necessarily have one client (and only one client) for whom all seven
forecasts were correct.
By following this procedure with several different groups of 128 clients, and
starting new groups each week, the firm may be able to generate enough positive
responses from clients for it to realize significant profits.
Guaranteed Winners
There is another scheme that is somewhat related to the one just described but that is
even more elegant because of its simplicity. In this scheme, a firm advertises that for
a fixed fee, usually 10 or 20 dollars, it will send the client its forecast of the winner of
any upcoming baseball game, football game, boxing match, or other sports event that
the client might specify. Furthermore, the firm offers a money-back guarantee that
this forecast will be correct; that is, if the team or person designated as the winner in
the forecast does not actually turn out to be the winner, the firm will return the full
fee to the client.
How should you react to such an advertisement? At first glance, it would appear
that the firm must have some special knowledge about these sports events, because
otherwise it could not afford to guarantee its forecasts. Further reflection reveals,
however, that the firm simply cannot lose, because its only expenses are those for
advertising and postage. In effect, when this scheme is used, the firm holds the client’s
fee until the winner has been decided. If the forecast was correct, the firm keeps the
fee; otherwise, it simply returns the fee to the client.
On the other hand, the client can very well lose. He presumably purchases the
firm’s forecast because he desires to bet on the sports event. If the forecast proves to
be wrong, the client will not have to pay any fee to the firm, but he will have lost any
money that he bet on the predicted winner.
Thus, when there are “guaranteed winners,” only the firm is guaranteed to win.
In fact, the firm knows that it will be able to keep the fees from all the clients for
whom the forecasts were correct.
1.12 Supplementary Exercises 53
Improving Your Lottery Chances
State lotteries have become very popular in America. People spend millions of
dollars each week to purchase tickets with very small chances of winning medium
to enormous prizes. With so much money being spent on lottery tickets, it should not
be surprising that a few enterprising individuals have concocted schemes to cash in
on the probabilistic na¨ıvete of the ticket-buying public. There are now several books ´
and videos available that claim to help lottery players improve their performance.
People actually pay money for these items. Some of the advice is just common sense,
but some of it is misleading and plays on subtle misconceptions about probability.
For concreteness, suppose that we have a game in which there are 40 balls numbered 1 to 40 and six are drawn without replacement to determine the winning
combination. A ticket purchase requires the customer to choose six different numbers from 1 to 40 and pay a fee. This game has 40
6
= 3,838,380 different winning
combinations and the same number of possible tickets. One piece of advice often
found in published lottery aids is not to choose the six numbers on your ticket too far
apart. Many people tend to pick their six numbers uniformly spread out from 1 to 40,
but the winning combination often has two consecutive numbers or at least two numbers very close together. Some of these “advisors” recommend that, since it is more
likely that there will be numbers close together, players should bunch some of their
six numbers close together. Such advice might make sense in order to avoid choosing
the same numbers as other players in a parimutuel game (i.e., a game in which all
winners share the jackpot). But the idea that any strategy can improve your chances
of winning is misleading.
To see why this advice is misleading, let E be the event that the winning combination contains at least one pair of consecutive numbers. The reader can calculate
Pr(E) in Exercise 13 in Sec. 1.12. For this example, Pr(E) = 0.577. So the lottery aids
are correct that E has high probability. However, by claiming that choosing a ticket in
E increases your chance of winning, they confuse the probability of the event E with
the probability of each outcome in E. If you choose the ticket (5, 7, 14, 23, 24, 38),
your probability of winning is only 1/3,828,380, just as it would be if you chose any
other ticket. The fact that this ticket happens to be in E doesn’t make your probability of winning equal to 0.577. The reason that Pr(E) is so big is that so many different
combinations are in E. Each of those combinations still has probability 1/3,828,380
of winning, and you only get one combination on each ticket. The fact that there are
so many combinations in E does not make each one any more likely than anything
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