Number Theoretical Ciphers - Modern Cryptanalysis: Techniques for Advanced Code Breaking - page 22

2.1.2 Dependence
Probabilities can be more complicated, as mentioned above. For example, sometimes events are not so easily
modeled as a set that adds up to 1. For example, say we are rolling a six-sided die, and we define a simple
game: Alice scores if the number that comes up is even, and Bob scores if the number is prime (see Figure 2-1 ) .
Whoever scores wins, except if both or neither scores, in which case it is a draw. This means that in the sticky
situation when we roll a 2, which is both even and prime, both score, and therefore it is a draw. In this case, if
Alice scores, then that affects the probability of Bob scoring — Alice scores on a 2, 4, or 6, which means that
Bob now has a 1/3 chance of scoring instead of the 1/2 chance he had before (since Bob scores on a 2, 3, or 5).
Figure 2-1 The Alice and Bob dice game. Here, A = 1 if Alice scores, and B = 1 if Bob scores.
The preceding example can be broken into a normal set of occurrences of the die rolling a number in the set
{1, 2, 3, 4, 5, 6}, and the above rules constituting subsets of this, so that the probabilities will again add up to 1.
(For example, we would have the rolls {1, 2, 3, 4, 5, 6} represent { D, D, B, A, B, A }, where D means draw, B
means Bob wins, and A means Alice wins.) The particular situation we are analyzing might dictate one repres-
entation or another to use (such as if we are interested in who wins or who scores).
Measuring the probability of a draw, which in the above happens when a 2 is thrown on the die, is more
complicated because the events are dependent — the outcome of one event happening affects the probability of
another event happening. If events are independent of one another, then one happening will have no influence
on the probabilities in another event. For instance, if we threw the die a second time, the second throw would
have the same probabilities again, since they are not affected by the previous throw. However, if we threw four
dice, calculating how many times a 6 was rolled, and rerolled any dice that did not roll a 6 a second time, then
the outcome of the first roll will affect the second one — the throwing of the dice would then no longer be in-
dependent.
2.1.2.1 Fun with Poker 1
Maybe a more concrete example or two might solidify a few of the above ideas — at least, more than the pre-
ceding, slightly abstract examples.
Poker is a game played with a standard 52-deck of playing cards (see Figure 2-2 ). There are four suits
(shapes associated with particular cards), each having an equal number of 13 cards: (in increasing order of
power) numbered cards 2 through 10, a Jack, a Queen, a King, and an Ace (which doubles as a virtual 1 as
well).