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# Introduction to Probability ## Terminology ### Experiment An activity with observable results. ### Sample Space The set of all possible outcomes of an experiment. ### Event A subset of the sample space. ## Example Experiment: Flip a coin Sample Space: {$H, T$} Event: Getting a head {$H$...

# Introduction to Probability ## Terminology ### Experiment An activity with observable results. ### Sample Space The set of all possible outcomes of an experiment. ### Event A subset of the sample space. ## Example Experiment: Flip a coin Sample Space: {$H, T$} Event: Getting a head {$H$} ## Probability ### Definition If an experiment has $n$ equally likely outcomes and an event $A$ occurs in $m$ of these outcomes, then the probability of $A$ is: $P(A) = \frac{m}{n} = \frac{\text{number of outcomes in A}}{\text{total number of outcomes}}$ ### Properties of Probability 1. $0 \leq P(A) \leq 1$ 2. $P(S) = 1$, where $S$ is the sample space 3. $P(\phi) = 0$, where $\phi$ is the null set ### Example Experiment: Roll a die Sample Space: ${1, 2, 3, 4, 5, 6}$ Event $A$: Roll an even number ${2, 4, 6}$ $P(A) = \frac{3}{6} = \frac{1}{2}$ ## Union and Intersection ### Union The union of two events $A$ and $B$, denoted by $A \cup B$, is the event containing all outcomes that are in $A$ or $B$ or both. ### Intersection The intersection of two events $A$ and $B$, denoted by $A \cap B$, is the event containing all outcomes that are in both $A$ and $B$. ### Mutually Exclusive Events Two events $A$ and $B$ are mutually exclusive if $A \cap B = \phi$, i.e., they have no outcomes in common. ## Addition Rule $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ If $A$ and $B$ are mutually exclusive, then $P(A \cap B) = 0$, and the rule simplifies to: $P(A \cup B) = P(A) + P(B)$ ## Complement The complement of an event $A$, denoted by $A'$, is the set of all outcomes in the sample space that are not in $A$. ### Complement Rule $P(A') = 1 - P(A)$ ## Conditional Probability The conditional probability of event $A$ given that event $B$ has occurred is: $P(A|B) = \frac{P(A \cap B)}{P(B)}$, provided $P(B) > 0$ ## Independent Events Two events $A$ and $B$ are independent if the occurrence of one does not affect the probability of the other. ### Test for Independence $P(A|B) = P(A)$ or $P(B|A) = P(B)$ or $P(A \cap B) = P(A)P(B)$ ## Multiplication Rule $P(A \cap B) = P(A|B)P(B)$ or $P(A \cap B) = P(B|A)P(A)$ If $A$ and $B$ are independent, then $P(A \cap B) = P(A)P(B)$