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Probability

tossed in a random manner, it can land heads or tails, but we do not know which of these will occur in a single

toss. However, suppose we let s be the number of times heads appears when the coin is tossed n times. As n

increases, the ratio f   = s/n, called the relative frequency of the outcome, becomes more stable. If the coin is

perfectly balanced, then we expect that the coin will land heads approximately 50% of the time or, in other words,

the relative frequency will approach¹2. Alternatively, assuming the coin is perfectly balanced, we can arrive at

the value¹2 deductively. That is, any side of the coin is as likely to occur as the other; hence the chance of getting

a head is 1 in 2 which means the probability of getting heads is¹2 . Although the speci?c outcome on any one toss

is unknown, the behavior over the long run is determined. This stable long-run behavior of random phenomena

forms the basis of probability theory.

A probabilistic mathematical model of random phenomena is de?ned by assigning “probabilities” to all the

possible outcomes of an experiment. The reliability of our mathematical model for a given experiment depends

upon the closeness of the assigned probabilities to the actual limiting relative frequencies. This then gives rise

to problems of testing and reliability, which form the subject matter of statistics and which lie beyond the scope

of this text.

7.2    SAMPLE SPACE AND EVENTS

The set S of all possible outcomes of a given experiment is called the sample space. A particular outcome,

i.e., an element in S, is called a sample point. An event A is a set of outcomes or, in other words, a subset of the

sample space S. In particular, the set {a } consisting of a single sample point a ? S is called an elementary event.

Furthermore, the empty set     and S itself are subsets of S and so    and S are also events;    is sometimes called

the impossible event or the null event.

Since an event is a set, we can combine events to form new events using the various set operations:

(i)   A ? B is the event that occurs iff A occurs or B occurs (or both).

(ii)   A ? B is the event that occurs iff A occurs and B occurs.

(iii)    Ac, the complement of A, also written &, is the event that occurs iff A does not occur.

Two events A and B are called mutually exclusive if they are disjoint, that is, if A? B = . In other words, A

and B are mutually exclusive iff they cannot occur simultaneously. Three or more events are mutually exclusive

if every two of them are mutually exclusive.