The citizen., 1908-01-10
The TOSS OF A CON-,
Mathematics of the Turning of
Heads or Tails.
CHANCE AND THEORY CLASH.

IF Heads Turn Ten Times In Sequence,
Theory Says the Odds Are Against
Another Head, Yet Chance Says the
Odds on the Next Toss Are Even.
A famous mathematician, Professor
Karl Pearson, once spent the greater
part of his vacation deliberately toss
ing Il shilling and making careful notes
of how it fell. He spun the shilling
2o,O00 times and pupil of his, work
ing separately, spun penny 5000
times and also tested the drawing of
1900 tickets from a bag.
1t may seem strange that learned
professor should put himself to such
an amount of trouble to demonstrate
what every schoolboy who had ever
tossed coin already knew. Yet, as
matter of fact. few really do grasp
the laws which govern such an appar-
emily straightforward matter as the
tossing of coin In the words of the
arithmetician. the theory of runs
that Is, heads turning up repeatedly
or tails turning up repeatedly-is pre-
cisely as follows:
The chance of head is one-half; of
two heads following is one-half multi
piled by one-half-that Is. one-quarter:
of three heads in succession, one-half
multiplied by one-half multiplied by
one-half-that is, one-eighth. Now,
what do you suppose is the chance of
run of eleven heads? 1t is safe to
say that not many persons, however
accustomed to tossinFcoins. have rea-
soned this out. TFefaet is that one
run of eleven heaus is on the aver
age only to be expected in 2,04S sets
of coin tossing.
Although the man In the street may
not have reckoned this. he is always
quite positive that if, say. a coin has
fallen ten times head upward he is
safe to start backing tails. He puts
his money on tails turning up be-
cause, he says. it stands to sense that
the run of heads can't continue But
does it? At the eleventh toss the head
of the coin is just as big as it ever
was. What mysterious influence can
past event, the tossing Of ten heads,
have on future one which has no
link with them-namely. the tossing
of the coin the eleventh time? Surely
each toss is an event by itself, as Sir
Hiram Maxim said of a game Mt rou-
lette at Monte Carlo:
at is a pure. unadulterated question
0f oI,ouoa opd n iS At iuG,,apnoJ ~,
the least by anything which has ever
taken place before or that ever will
take place in the future"
A nasty piece of plain speaking this
for the cranks who had published
schemes for breaking the bank and
whose plans depended entirely on the
theory that If one game ended in a
win for "red" the chances against it
ending ''red" a second time were less,
third time less still, and so on.
This of course would be sound
enough argument provided that you
regard some dozens of games of rou-
lette or tosses of coin all as one con-
tinuous event 1t is quite safe. for in
stance, to offer beforehand big odds
against a coin turning up heads ten
times running. But in practice the
public house loafer does not do this.
What he does is to bet on each Sepa-
rate toss by itself, thus defeating his
own aims. The odds against coin
turning up heads eleven times are as
has been shown, something like 1000
to l. But suppose you only start bet
! ting at the tenth toss What are the
odds against the eleventh toss again
ling head?
Ne odds. so far from being 1000 to
actually l to 1! To use an
the odds are even-that is
\Vou split up the eleven tosses
separate events to be bet
your bets should be
Aall the time, however
up running. But if
y tosses as one COM
v offer n prelimi-
whole eleven re-
will have to
that
kties