# 2. Calculating Chances¶

Once you start working with probabilities, you quickly realize that the assumption of all possible outcomes being equally likely isn’t always reasonable. For example, if you think a coin is biased then you won’t want to assume that it lands heads with the same chance as tails.

To deal with settings in which some outcomes have a higher chance than others, a more general theory is needed. In the 1930’s, the Russian mathematician Andrey Kolmogorov (1903-1987) formulated some ground rules, known as axioms, that covered a rich array of settings and became the foundation of modern probability theory.

The axioms start out with an outcome space $$\Omega$$. We will assume $$\Omega$$ to be finite for now. Probability is a function $$P$$ defined on events, which as you know are subsets of $$\Omega$$. The first two axioms just set the scale of measurement: they define probabilites to be numbers between 0 and 1.

• Probabilities are non-negative: for each event $$A$$, $$P(A) \ge 0$$.

• The probability of the whole space is 1: $$P(\Omega ) = 1$$.

The third and final axiom is the key to probability being a “measure” of an event. We will study it after we have developed some relevant terminology.