AND and OR with Overlap or Collisions

We can now go one step further and consider the situation where events A and B are not mutually exclusive (that is, A and B might occur together sometimes or perhaps overlap in some way). Figure 2-1 illustrates the case. As an example, let's continue to observe pairs of dice, but the experiment will be to toss three die at once. All die remain independent, but the occurrence of the event "7" or "5" is no longer mutually exclusive. The event {3,4,1} might be rolled providing the opportunity for the (3,4) pair of the "7" event and the (4,1) pair of the "5" event. However, we are looking for p(A) or p(B) but not p(A * B); therefore, those tosses like event {3,4,1} where "A and B" occur cannot be counted, thereby reducing the opportunities for either A or B. Throwing out the occurrence of "A and B" reduces the chances for "A or B" alone to happen. From this reasoning comes a more general equation for OR:

Figure 2-1: Overlapping Events.

Notice that if A and B are mutually exclusive, then p(A * B) = 0, which gives a result consistent with the earlier equation given for the OR situation.

Looking at the above equation, the savvy project manager recognizes that risk has increased for achieving a successful outcome of either A or B since the possibility of their joint occurrence, overlap, or collision of A and B takes away from the sum of p(A) + p(B). Such a situation could come up in the case of two resources providing inputs to the project, but if they are provided together, then they are not useful. Such collisions or race conditions (a term from system engineering referring to probabilistic interference) are common in many technology projects.