May 30

Basic probability definitions (MECE, outcome vs. event)

by David Harper, CFA, FRM, CIPM

The first learning outcome in the 2007 financial risk manager (FRM) list is define the following terms:

Random variable (or stochastic variable): function that defines a point in the sample space of outcomes.

Outcome: the result of a single trial.

Event: the result that reflects none, one, or more outcomes in the sample space. Events can be simple or compound.

Mutually exclusive events: events which cannot simultaneously occur. If A and B are mutually exclusive, the probability of (A and B) is zero. Put another way, their intersection is the null set.

Exhaustive events: events that cumulatively describe all possible outcomes.

If you roll a single six-sided die, there are six possible outcomes. An event is user-defined. It could be, "what is the probability that I roll an even number?" Three outcomes (2,4, or 6) produce the single event (rolling even). If you play craps, an event might be "rolling a seven with two dice." There are six outcomes that produce the event; notice that both 1+6 and 6+1 are separate events.

Mutually exclusive and cumulatively exhaustive (MECE) combine to form a helpful concept. Let's say the total population of people-icons, for argument's sake, included (i) the teacher, (ii) the policewoman, (iii) the artist, (iv) the magician, and (v) superman.

This set is mutually exclusive but not cumulatively exhaustive because it doesn't include everybody: