H = and probability of T = . We find that both the assignments ( ) and ( ) are valid for probability of H and T. In fact, we can assign the numbers p and ( – p ) to both the outcomes such that ≤ p ≤ and P(H) + P(T) = p + ( – p ) = This assignment, too, satisfies both conditions of the axiomatic approach of probability. Hence, we can say that there are many ways (rather infinite) to assign probabilities to outcomes of an experiment.
We now consider some examples. Example Let a sample space be S = { ω , ω ,..., ω }.Which of the following assignments of probabilities to each outcome are valid? Outcomes ω ω ω ω ω ω (a) (b) (c) (d) (e) . .
Solution (a) Condition (i): Each of the number p( ω i ) is positive and less than one. Condition (ii): Sum of probabilities MATHEMATICS Therefore, the assignment is valid (b) Condition (i): Each of the number p ( ω i ) is either or . Condition (ii) Sum of the probabilities = + + + + + = Therefore, the assignment is valid (c) Condition (i) Two of the probabilities p ( ω ) and p ( ω ) are negative, the assignment is not valid (d) Since p ( ω ) = > , the assignment is not valid (e) Since, sum of probabilities = . + .
= . , the assignment is not valid. . .
Probability of an event Let S be a sample space associated with the experiment ‘examining three consecutive pens produced by a machine and classified as Good (non-defective) and bad (defective)’. We may get , , or defective pens as result of this examination. A sample space associated with this experiment is