Statistics for Maintenance Modelling: Basic Probability Concepts

# Key Definitions and Concepts¤

In order to understand the subjects studied in later sessions it will be beneficial to get to grips with the basic concepts of probability. Click on the concepts for a definition and examples.

- An Experiment
- The Sample Space
- An Event
- The Union of Events
- Intersection of Events
- The Complement of an Event
- An Event contained in Another Event

An activity for which the outcome is not predictable with certainty is an experiment. However, although the outcome of the experiment is uncertain, the set of all possible outcomes is known.

The set of all possible outcomes of an experiment is known as the sample space of the experiment, and is denoted S.

Some examples of sample spaces are:

1. If the experiment consists of flipping a coin once, then the sample space may be written as:
   S = {H,T} where H and T stand for heads and tails.
2. If the experiment consists of measuring the lifetime of a bulb (in hours) then the sample space may be written as:
   S = {x : 0 x < } where x represents the useful lifetime of the bulb.
3. If the experiment consists of tossing two dice simultaneously, then the sample space consists of the 36 points given by:
   S = {(i, j) : i, j = 1,2,3,4,5,6} where the outcome (i, j) is said to occur if i appears on one of the die, and j on the other.

Any subset of the sample space S is called an event, E. Some examples of events, for the sample spaces described earlier, are:

1. In the 1st sample space S = {H,T}, if E = H, then E is the event that a head was thrown.
2. In the 2nd sample space S = {x : 0 x < }, if E = {x: 0 x 5}, then E is the event that the light bulb does not last longer than 5 hours.
3. In the 3rd sample space S = {(i, j) : i, j = 1,2,3,4,5,6}, if E = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}, then E is the event that the sum of the dice is 7.

For any two events E and F of a sample space S, we define the new event EF to consist of all outcomes that are either in E, or in F, or in both. In other words, the event EF will occur if either E or F occurs.

In the 3rd sample space S = {(i, j) : i, j = 1,2,3,4,5,6}, if E = {(1,6), (2,5)} and
F = {(3,4)}, then EF = {(1,6), (2,5), (3,4)}.

The event EF is called the union of event E and the event F.

For any two events E and F the intersection of E and F, written EF, consists of all outcomes that are both in E and in F.

In the 2nd sample space S = {x : 0 x < }, if E = {x: 0 x 6} and
F = {x: 4 x 15}, then EF = {x: 4 x 6}.

For an event E, the definition of the complement of E, written E is the set of all outcomes in the sample space not equal to E. In other words E will occur if E does not.

For instance, in 3rd sample space S = {(i, j) : i, j = 1,2,3,4,5,6}, if event
E = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}, then E = {the event that the dice do not sum to 7}.

Since either the event or its complement must occur, we have P(E) + P(E) = 1 i.e. the probability of the event E occuring plus the probability of it not occuring is 1.

Finally, if all the outcomes of event E are contained in event F, then we say that E is contained in F, written EF (or FE). This means that if EF, the occurrence of E necessarily implies the occurrence of F.