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Discussion 7B

Probability

sample space: $\Omega$ , set of all possible outcomes, or sample points $\omega$

event: a subset of the sample space

Probability is an assignment of a number to each sample point such that

$0 \leq \Pr[\omega] \leq 1$

$\sum_{\omega \in \Omega} \Pr[\omega] = 1$

Since sample points are disjoint, the probability of an event $A$ occurring is the sum of the probabilities of $A$ ’s sample points

If all sample points have equal probability of occurring, then $\Pr[\omega] = \frac{1}{|\Omega|}$

Then the probability of an event $A$ occurring is $\Pr[A] = \sum_{\omega \in A} \Pr[\omega] = \frac{|A|}{|\Omega|}$

This is just a counting problem: find $|A|$ and $|\Omega|$

The probability of a sequence of independent events (independence next time) $A_1, A_2, \cdots, A_n$ is the product of the probabilities of the individual events

Sometimes it is easier consider the complement of a certain event

$|\bar{A}| + |A| = |\Omega|$

$\Pr[\bar{A}] + \Pr[A] = 1$

This generalizes to any partitioning of $\Omega$ : a list of subsets that are disjoint and for which their union is $\Omega$