Page 201 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes The simplest manner to specify a probability P on a discrete sample space is to state first the
probability pi of every elementary event {ai} and then to define
P(A) = _ i:ai  A

pi , for all A  .
This thought is graphically displayed in Figure 14.10. Each element ai in the sample is allocated
a probability weight pi illustrated by a black dot.

Did u know? To discover the probability of the set A we have to sum up the weights of all
the essentials in A.

Figure 14.10: A Discrete Sample Space

Again, it is up to the modeller to correctly state these probabilities. Luckily, in many applications
all basic events are equally likely, and therefore the probability of every basic event is equal to 1
divided by the total number of elements in .

Since the “equally likely” principle is so significant, we invent it as a theorem.
Theorem: (Equilikely Principle): If  has a finite number of results, and all are uniformly likely,
then the probability of each event A is defined as
P(A) = |A|
||.

So for such sample spaces the computation of probabilities diminishes to counting the number of
outcomes (in A and ).
When the sample space is not countable, for example  = R+, it is supposed to be continuous.

Example: We draw at random a point in the interval [0, 1]. Every point is equally
probable to be drawn. How do we state the model for this experiment?
The sample space is clearly  = [0, 1], which is a continuous sample space. We cannot define P by
means of the simple events {x}, x  [0, 1] since each of these events must have probability 0 (!).
Though we can define P as follows:

For each 0  a  b  1, let P([a, b]) = b – a .

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