Concrete mathematics : a foundation for computer science

368 DISCRETE PROBABILITY
total of 6’ = 36 elements.
We usually assume that dice are “fair,” namely that each of the six possibilities
for a particular die has probability i, and that each of the 36 possible
rolls in n has probability 8. But we can also consider “loaded” dice in which Careful: They
there is a different distribution of probabilities. For example, let
might go off.
Prl(m) = Pr,(m) = +;
Prl(a) = Prl(m) = Prj(m) = Prl(m) = f.
Then LED Prl (d) = 1, so Prl is a probability distribution on the set D, and
we can assign probabilities to the elements of f2 = D2 by the rule
Pr,,(dd’) = Prl(d) Prl(d’). (8.2)
For example, Prlj ( � m) = i. i = A. This is a valid distribution because
x Prll(w) = t Prll(dd’) = t Prl(d) Prl(d’)
wen dd’EDZ d,d’ED
= x Prl(d) x Prr(d’) = 1 . 1 = 1 .
dED
d’ED
We can also consider the case of one fair die and one loaded die,
Prol(dd’) = Pro(d) Prl(d’), where Pro(d) = 5, (8.3)
in which case ProI ( � m) = i . i = &. Dice in the “real world” can’t really
be expected to turn up equally often on each side, because there is not perfect If all sides of a cube
symmetry; but i is usually pretty close to the truth.
were identical, how
could we tell which
An event is a subset of n. In dice games, for example, the set
side is face up?
is the event that “doubles are thrown!’ The individual elements w of 0 are
called elementary events because they cannot be decomposed into smaller
subsets; we can think of co as a one-element event {w}.
The probability of an event A is defined by the formula
Pr(wE A) = x Pr(w);
WEA
(8.4)
and in general if R(o) is any statement about w, we write ‘Pr(R(w))’ for the
sum of all Pr(w) such that R(w) is true. Thus, for example, the probability of
doubles with fair dice is $ + & + & + $ + $ + & = i; but when both dice are
loaded with probability distribution Prl it is 1+~+~+~+~+~ = & > i.
16 64 64 64 64 16
Loading the dice makes the event “doubles are thrown” more probable.