1.1 Introduction
- Probability is a mathematical language for quantifying uncertatinty
1.2 Sample Spaces and Events
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Sample Space $\Omega$ : is the set of possible outcomes of an experiment
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Events : Subsets of Ω are called Events
1.3 Probability
- A function $\mathbb{P}$ that assigns a real number $ \mathbb{P}(A) $ to every event $A$ is a probability distribution or a probability measure.
1.4 Probability on Finite Sample Spaces
- If $\Omega$ is finite and if each outcome is equally likely, then, the uniform probability distribution is
$$ \mathbb{P} = \frac{|A|}{| \Omega |}$$
- And we need to count the number of points in an event using combinational methods, to compute probabilities.
$${n \choose x} = \frac {n \times (n-1) \times … (n-k-1)}{k !} $$
1.5 Independent Event
- Two events $A$ and $B$ are independent if $ \mathbb{P}(AB) = \mathbb{P}(A) \mathbb{P}(B)$
1.6 Conditional probability
- If $\mathbb{P}(B)>0$ then the conditional probability of A given B is
$$ \mathbb{P(A|B)} = \frac{\mathbb{P}(AB)}{\mathbb{P}(B)} = \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} $$
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In general, $ \mathbb{P} (A | B) \neq \mathbb{P} (B | A) $
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$A$ and $B$ are independent if and only if $ \mathbb{P}(A | B) = \mathbb{P}(A) $
1.7 Bayes’ Theorem
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Basis of expert systems and Bayes’ nets
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Bayes’ Theorem : Let $A_1, … A_k$ be a partition of $\Omega$ such that $\mathbb{P}(A_i) > 0$ for each $i$. If $\mathbb{P}(B)>0$, then, for each $i = 1 , … k$,
$$ \mathbb{P}(A_i | B) = \frac{\mathbb{P}(A_i B)}{\mathbb{P}(B)} = \frac{\mathbb{P}(B | A_i) \mathbb{P}(A_i)}{\mathbb{P}(B)} = \frac{\mathbb{P}(B | A_i) \mathbb{P}(A_i)}{ \sum_j \mathbb{P}(B | A_j) \mathbb{P}(A_j)} $$