318 6 Combinatorics and Probability
already five propeller planes and one jet plane. Later, a farmer sees a jet plane flying
out of Eulerville. What is the probability that the plane that arrived from Gauss
City was a propeller plane, provided that all events are equiprobable?924.A coin is tossedntimes. What is the probability that two heads will turn up in
succession somewhere in the sequence?
925.Two people,AandB, play a game in which the probability thatAwins isp, the
probability thatBwins isq, and the probability of a draw isr. At the beginning,
Ahasmdollars andBhasndollars. At the end of each game, the winner takes a
dollar from the loser. IfAandBagree to play until one of them loses all his/her
money, what is the probability ofAwinning all the money?
926.We play the coin tossing game in which if tosses match, I get both coins; if they
differ, you get both. You havemcoins, I haven. What is the expected length of
the game (i.e., the number of tosses until one of us is wiped out)?
6.3.3 Geometric Probabilities
In this section we look at experiments whose possible outcomes are parametrized by the
points of a geometric region. Here we interpret “at random’’ to mean that the probability
that a point lies in a certain region is proportional to the area or volume of the region. The
probability of a certain event is then computed by taking the ratio of the area (volume)
of the favorable region to the area (volume) of the total region. We start with the game
of franc-carreau investigated by George-Louis Leclerc, Comte de Buffon, in his famous
Essai d’arithmétique morale.
Example.A coin of diameterdis thrown randomly on a floor tiled with squares of side
l. Two players bet that the coin will land on exactly one, respectively, more than one,
square. What relation shouldlanddsatisfy for the game to be fair?
Figure 42