Probability Space

#Math

• A probability space is a triple of $(\Omega,\mathcal{F},P)$:
  • $\Omega$ is the set of all possible outcomes of some random experiment
  • $\mathcal{F}$ is the power set of $\Omega$ (all possible events you could try to ask the probability of)
  • $P$ is a probability measure
  • An event $A$ satisfies: $A\in\mathcal{F}$ and $A\subseteq\Omega$

• E.g.: What is the probability of getting at least 1 head if you flipped two coins?
  • $\Omega={HH,HT,TH,TT}$
  • $\mathcal{F}=\mathcal{P}(\Omega)$
• Let $A$ the event that you get at least 1 head
  • $A={HH,HT,TH}$
  • Then of course, $A\in\mathcal{F}$ and $A\subseteq\Omega$
  • $P(A)=\frac{|A|}{|\Omega|}=\frac{3}{4}$