312 . ANNUAL REPORT SMITHSONIAN INSTITUTION, 1959 
the game can go on indefinitely, the lead tends not to change hands 
nearly so often as most people would guess. In a game where both 
players have an equal chance of winning—such as matching pennies— 
after 20,000 plays it is about 88 times as likely that the winner has led 
all the time as that the two players have shared the lead equally. No 
matter how long the games lasts, it is more likely that one player has 
led from the beginning than that the lead has changed hands any 
given number of times, 
The random-walk abstraction is applicable to a great many physical 
situations. Some clearly involve chance—e.g., diffusion of gases, flow 
of automobile traffic, spread of rumors, progress of epidemic disease. 
The technique has even been applied to show that after the last glacial 
period seed-carrying birds must have helped reestablish the oak forests 
in the northern parts of the British Isles. But some modern random- 
walk problems have no obvious connection with chance. In acompli- 
cated electrical network, for example, if the voltages at the terminals 
are fixed, the voltages at various points inside the circuit can be 
calculated by treating the whole circuit as a sort of two-dimensional 
gambler’s ruin game. 
RISK VERSUS GAIN 
Mathematical statistics, the principal offshoot of probability theory, 
is changing just as radically as probability theory itself. Classical 
statistics has acted mainly as a tribunal, warning users against draw- 
ing risky conclusions. The judgments as handed down are always 
somewhat equivocal, such as: “It is 98 percent certain that drug A is 
at least twice as potent as drug B.” But what if drug A is actually 
only half as potent? Classical statistics admits this possibility, but 
does not evaluate the consequences. Modern statisticians have gone a 
step farther with a new set of ideas known collectively as decision 
theory. “We now try to provide a guide to actions that must be taken 
under conditions of uncertainty,” explains Herbert Robbins of Colum- 
bia. “The aim is to minimize the loss due to our ignorance of the true 
state of nature. In fact, from the viewpoint of game theory, statisti- 
cal inference becomes the best strategy for playing the game called 
science.” 
The new approach is illustrated by the following example. A 
philanthropist offers to flip a coin once and let you call “heads” or 
“tails.” If you guess right, he will pay you $100. You notice the 
coin is so badly bent and battered that it is much more likely to land 
on one side than the other. But you cannot decide which side the 
coin favors. The philanthropist is willing to let you test the coin with 
trial flips, but he insists you pay him $1 for each experiment. How 
many trial flips should you buy before you make up your mind? The 
answer, of course, depends on how the trials turn out. If the coin 
