Consider a probability function defined on a sample space

MATHEMATICS · Subjective · A Level · Mathematics · Probability · Probability

Problem

Consider a probability function

P

defined on a sample space

S

. (i) Using the axioms of probability, prove that the probability of the empty set is zero, i.e.

P(\emptyset)=0

. (ii) Show that for any event

E\subseteq S

, the inequality

0\le P(E)\le 1

holds. (iii) Prove that

P(E^{c})=1-P(E)

, where

E^{c}

is the complement of

E

S

. (iv) Deduce from the above results whether the event

S

itself is sure or impossible, giving appropriate reasoning.