Let be a sample space and suppose we have defined a set function, P(), which satisfies the properties P1P3 of Definition 1.10, on that

Let Ω be a sample space and suppose we have defined a set function, P(⋅), which satisfies the properties P1–P3 of Definition 1.10, on that space. Examine whether each of the set functions defined below can be used as a probability on that space;

here, A is an arbitrary event on Ω.

(i) P1(A) = P(A)∕2 (ii) P2(A) = 1 − P(A)

(iii) P3(A) = |P(A)| (iv) P4(A) = [P(A)]2

(v) P5(A) = (P(A) + 1)∕2 (vi)P6(A) = (1 − P(A))∕2.