Modern Engineering Thermodynamics

18.7 Introduction to Mathematical Probability 743
WHAT ARE THE GAMES CRAPS AND DICE?
Craps is a simplified version of the old game of hazard introduced in Europe during the Crusades. Craps is played between
two players. If the first throw is a 7 or 11 (a natural), the thrower wins immediately. If it a 2, 3, or 12 (craps, which was
called crabs in the old game of hazard), the thrower loses immediately. If any other number is thrown the thrower goes on
throwing until either the same number or a 7 occurs. If a 7 comes up before the first number thrown, the thrower loses;
otherwise, the thrower wins.
Most casino dice are custom-made of celluloid with a tolerance of one 10,000th of an inch (even the heat of a shootor’s hands
may slightly alter the size of a pair of dice). The celluloid is “cured” over a period of time to dry it out and give it stability.
Then, it goes through a series of milling operations to form it into a perfect cube. The shallow indentations that form the spots
are made on all six sides. These indentations are filled with a polyester resin, and the dice are then subjected to a final grinding,
which leaves them with smooth surfaces. Cheap plastic dice are polished in a mechanical tumbler similar to those used by lapidarists,
but such mechanical polishing also gives them rounded edges which can affect the outcome of a throw. Some casinos
have dice made in odd sizes to prevent a dishonest player from switching loaded dice for honest dice. Dice in most casinos are
changed every 30 days to maintain precise sizing, and pairs of used dice are often given to customers as souvenirs.
This proportionality can be reduced to an equality by the introduction of a proportionality constant P c ,which
we call the collision probability. Then, the number of collisions that occur in time interval Δt becomes
In the limit as Δt → 0, this equation becomes
δN = −P c NV avg ðΔtÞ
dN = −P c NV avg dt
Let V avg dt = dX, where X is the distance between collisions. This equation can be written as
which can be integrated to give
dN
N = −P c dX
N = NðtÞ = N 0 expð−P c XÞ (18.31)
where N 0 = N(t = 0). In Eq. (18.15), we define the mean free path λ as the average distance between collisions,
which can be determined from X as
λ = X avg =
1 Z ∞
Xð−dNÞ
N 0
where the minus sign has been introduced because N(t) decreases with time, and therefore, dN < 0. From
Eq. (18.31), we find that
and the mean free path is given by
λ =
dN = −N 0 P c expð−P c XÞdX
1
Z ∞
XN 0 P c expð−P c XÞdX = 1 N 0 P c
0
Consequently, we have the result that the collision probability is exactly equal to the inverse of the mean free
path:
0
P c = 1 λ = σNV
where σ is the collision cross-section discussed earlier. Finally, Eq. (18.31) takes the form
NðtÞ = N 0 expð−X=λÞ
so that, by the time all the molecules in the gas have traveled a distance of only one mean free path (X = λ),
of them have not yet had a collision.
N
N 0
= e −1 = 0:368 = 36:8%