Definition 4: The axiomatic approach

None of the above definitions is entirely satisfactory or applicable for all situations. The Russian scientist Kolmogorov, therefore, proposed that probability should be defined axiomatically by stating three necessary and sufficient axioms (assumptions/properties):

1. All probabilities are greater than or equal to zero: $\Pr(A_i)\geq 0$ for all events $\{A_i\}$ (i.e. no event is more unlikely than a zero probability event).

2. The probabilities of all events in event space always sum up to one (i.e. something must happen !).

3. The probability of either one or other mutually exclusive events (i.e. events that cannot happen at the same time) is equal to the sum of the probabilities of each event alone. In other words, Pr($A_1$ or $A_2$)=Pr($A_1$)+Pr($A_2$) for all mutually exclusive events $A_1$ and $A_2$.

All the previous definitions satisfy these axioms and so provide valid and complementary interpretations of probability.