Mathematical_Recreations-Kraitchik-2e

1~6 :b-f athematical Recreations
go from player to player according to the rules given
above.
The sum of the probabilities given above is 1, which deJ
notes certainty. That would seem to say that if the calcu~
lations are correct any play is bound to end. This is not quite
the proper interpretation.. Suppose a 4 is thrown. What is
the probability that the player will never make his point?
This question is essentially meaningless. We have no experience
which would lead us to suppose that the thrower would
live more than 100 years to carryon the attempt, ann 100
years is a short time in the history of the earth. Instead, let
us see what is the probability that the play would continue
without issue for a large number of throws, say 100. The
probability that neither 4 nor 7 will turn up in one throw is
1 - 3 3~ 6 =~. The probability that neither will turn up in
100 consecutive throws is then Ct)l00 = 0.0000000000003, approximately.
Thus it is not very likely that the play would
go on without result for even an hour. In general, the probability
that the attempt to make the point will go on without
result for n throws is (t)n, a quantity which steadily decreases
as n increases, and which can be made as near to 0 as one
wishes by taking n sufficiently large. This is what we mean
by saying that the probability that the play will never ena
is 0, or that the play is certain to end.
6. THE GAME OF OVER TEN. This game is played with
three dice. The thrower wins if the total of the points turned
up in a single throw is over 10, and he loses if the total is Hi
or under.
An analysis similar. to that given for craps shows that the
various throws have the following frequencies:
Total. . . . 3 4 5 6 7 8 9 10
or. . . .. 18 17 16 15 14 13 12 11
Frequency 1 3 6 10 15 21 25 27