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Next: Conditional Probabilities Up: 2.1.1 Probability space Previous: Events Probabilities

Definition 3   Given a sample space $ \Omega$, a probability is an assignment of a number $ P$ to every event $ A \subseteq \Omega$ satisfying:
  1. $ 0 \le P(A) \le 1$ for all events $ A \subseteq \Omega$
  2. $ P(\Omega) = 1$
  3. if $ A_i \cap A_j = \phi \rightarrow P(A_i \cup A_j) = P(A_i) + P(A_j)$

Remark 1: If $ \Omega$ is discrete, then it is enough to specify $ P$ on each $ \omega \in \Omega$ to get $ P$ on each $ A \subseteq \Omega$. Just use (3) and add up:

$\displaystyle P(A) = \sum_{\omega \in A} P(\omega) $

Remark 2: In advanced probability the definition of a probability space is slightly more complicated, but not important here.

Birthdays.pdf is an handouts for the birthdays example.

Eran Borenstein