SOME BASIC DEFINITIONS OF PROBABILITY

Experiment:




Experiment is an activity that is either observed or measured, such as tossing a coin, or
drawing a card.



Event (Outcome):



An event is a possible outcome of an experiment. For example, if the experiment is to
sample six lamps coming off a production line, an event could be to get one defective and
five good ones.



Elementary Events:



Elementary events are those types of events that cannot be broken into other events. For
example, suppose that the experiment is to roll a die. The elementary events for this
experiment are to roll a 1 or a 2, and so on, i.e., there are six elementary events (1, 2,
3, 4, 5, 6). Note that rolling an even number is an event, but it is not an elementary
event, because the even number can be broken down further into events 2, 4, and 6.



Sample Space:




A sample space is a complete set of all events of an experiment. The sample space for the
roll of a single die is 1, 2, 3, 4, 5, and 6. The sample space of the experiment of
tossing a coin three times is:



First toss.........T T T T H H H H

Second toss.....T T H H T T H H

Third toss........T H T H T H T H



Sample space can aid in finding probabilities. However, using the sample space to express
probabilities is hard when the sample space is large. Hence, we usually use other
approaches to determine probability.



Unions & Intersections:




An element qualifies for the union of X, Y if it is in either X or Y or in both X
and Y. For example, if X=(2, 8, 14, 18) and Y=(4, 6, 8, 10, 12), then the union of
(X,Y)=(2, 4, 6, 8, 10, 12, 14, 18). The key word indicating the union of two or more
events is or.

An element qualifies for the intersection of X,Y if it is in both X and Y.
For example, if X=(2, 8, 14, 18) and Y=(4, 6, 8, 10, 12), then the intersection of
(X,Y)=8. The key word indicating the intersection of two or more events is and. See
the following figures:




Mutually Exclusive Events:



Those events that cannot happen together are called mutually exclusive events. For
example, in the toss of a single coin, the events of heads and tails are mutually
exclusive. The probability of two mutually exclusive events occurring at the same time is
zero. See the following figure:


Independent Events:



Two or more events are called independent events when the occurrence or nonoccurrence of
one of the events does not affect the occurrence or nonoccurrence of the others. Thus,
when two events are independent, the probability of attaining the second event is the same
regardless of the outcome of the first event. For example, the probability of tossing a
head is always 0.5, regardless of what was tossed previously. Note that in these types of
experiments, the events are independent if sampling is done with replacement.



Collectively Exhaustive Events:



A list of collectively exhaustive events contains all possible elementary events for an
experiment. For example, for the die-tossing experiment, the set of events consists of 1,
2, 3, 4, 5, and 6. The set is collectively exhaustive because it includes all possible
outcomes. Thus, all sample spaces are collectively exhaustive.



Complementary Events:




The complement of an event such as A consists of all events not included in A. For
example, if in rolling a die, event A is getting an odd number, the complement of A is
getting an even number. Thus, the complement of event A contains whatever portion of the
sample space that event A does not contain. See the following figure: