From Algorithms to Z-Scores - matloff - University of California, Davis

2.5. BASIC PROBABILITY COMPUTATIONS: ALOHA NETWORK EXAMPLE 11
P (X1 = 2) = P (C1 and C2 or not C1 and not C2)
(2.8)
= P (C1 and C2) + P ( not C1 and not C2) (from (2.2)) (2.9)
= P (C1)P (C2) + P ( not C1)P ( not C2) (from (2.4)) (2.10)
= p 2 + (1 − p) 2
(2.11)
(The underbraces in (2.8) do not represent some esoteric mathematical operation. There are there
simply to make the grouping clearer, corresponding to events G and H defined below.)
Here are the reasons for these steps:
(2.8): We listed the ways in which the event {X1 = 2} could occur.
(2.9): Write G = C1 and C2, H = D1 and D2, where Di = not Ci, i = 1,2. Then the events G and
H are clearly disjoint; if in a given line of our notebook there is a Yes for G, then definitely
there will be a No for H, and vice versa.
(2.10): The two nodes act physically independently of each other. Thus the events C1 and C2 are
stochastically independent, so we applied (2.4). Then we did the same for D1 and D2.
Now, what about P (X2 = 2)? Again, we break big events down into small events, in this case
according to the value of X1:
P (X2 = 2) = P (X1 = 0 and X2 = 2 or X1 = 1 and X2 = 2 or X1 = 2 and X2 = 2)
= P (X1 = 0 and X2 = 2) (2.12)
+ P (X1 = 1 and X2 = 2)
+ P (X1 = 2 and X2 = 2)
Since X1 cannot be 0, that first term, P (X1 = 0 and X2 = 2) is 0. To deal with the second term,
P (X1 = 1 and X2 = 2), we’ll use (2.5). Due to the time-sequential nature of our experiment here,
it is natural (but certainly not “mandated,” as we’ll often see situations to the contrary) to take A
and B to be {X1 = 1} and {X2 = 2}, respectively. So, we write
P (X1 = 1 and X2 = 2) = P (X1 = 1)P (X2 = 2|X1 = 1) (2.13)