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# Introduction to probability ## Terminology * **Experiment**: $E$, a process that produces an observation. * **Sample space**: $S$, the set of all possible outcomes of the experiment. * **Event**: $A$, a subset of the sample space. $\qquad$ e.g. $\qquad E$: Toss a coin twice. $\qquad S = \...

# Introduction to probability ## Terminology * **Experiment**: $E$, a process that produces an observation. * **Sample space**: $S$, the set of all possible outcomes of the experiment. * **Event**: $A$, a subset of the sample space. $\qquad$ e.g. $\qquad E$: Toss a coin twice. $\qquad S = \{HH, HT, TH, TT\}$ $\qquad A = \{HH, TT\}$ ## Probability A probability is a real number between 0 and 1. * $P(S) = 1$ * If $A$ and $B$ are mutually exclusive, then $P(A \cup B) = P(A) + P(B)$. From these axioms, we can derive other properties, such as: * $P(A^c) = 1 - P(A)$ * $P(\phi) = 0$ * If $A \subseteq B$, then $P(A) \le P(B)$ * $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ ## Equally Likely events If all outcomes are equally likely, then $P(A) = \frac{\text{number of outcomes in A}}{\text{number of outcomes in S}} = \frac{N(A)}{N(S)}$ **Example**: Two dice are thrown. What is the probability that the sum is 7? $S = \{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),$ $\qquad (2,1), (2,2), (2,3), (2,4), (2,5), (2,6),$ $\qquad (3,1), (3,2), (3,3), (3,4), (3,5), (3,6),$ $\qquad (4,1), (4,2), (4,3), (4,4), (4,5), (4,6),$ $\qquad (5,1), (5,2), (5,3), (5,4), (5,5), (5,6),$ $\qquad (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) \}$ $N(S) = 36$ $A = \{(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\}$ $N(A) = 6$ $P(A) = \frac{6}{36} = \frac{1}{6}$