Probability Theory PDF

# Probability ## Experiment & Sample Space - **Roll a die and observe the number.** - The **sample space** is the set of **all possible outcomes**: {1, 2, 3, 4, 5, 6} - **Flip a coin twice and observe the outcome.** - The **sample space** is the set of **all possible outcomes**: {HH,...

# Probability ## Experiment & Sample Space - **Roll a die and observe the number.** - The **sample space** is the set of **all possible outcomes**: {1, 2, 3, 4, 5, 6} - **Flip a coin twice and observe the outcome.** - The **sample space** is the set of **all possible outcomes**: {HH, HT, TH, TT} ## Probability of an Event - For every event A: $0 ≤ P(A) ≤ 1$ - $P(S) = 1$ - $P(Ø) = 0$ ### Mutually Exclusive Events - **If $A_1$ and $A_2$ are mutually exclusive (disjoint):** - $A_1$∩$A_2$ = Ø - $P(A_1 ∪ A_2) = P(A_1) + P(A_2)$ ### Not Mutually Exclusive Events - **If $A_1$ ⊆ $A_2$:** - $P(A_1) ≤ P(A_2)$ - $P(A_2 - A_1) = P(A_2) - P(A_1)$ - $A_2 - A_1 = A_2 ∩ A_1$ (The complement of the event $A_1$) - $P(A_1) = 1 - P(A_1)$ ## Two Events - **If A and B are two events:** - $P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$ - $P(A_1 ∪ A_2 ∪ A_3) = P(A_1) + P(A_2) + P(A_3) - P(A_1 ∩ A_2) - P(A_1 ∩ A_3) - P(A_2 ∩ A_3) + P(A_1 ∩ A_2 ∩ A_3)$ - When all outcomes are equally likely, $P(A) = P(A ∩ B) + P(A ∩ B') / p = n$ ## Three events - **A, B, and C events, then:** - **Distributive laws**: - $(A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C)$ - $(A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)$ - **De Morgan's laws**: - $(A ∪ B)' = A' ∩ B'$ - $(A ∩ B)' = A' ∪ B'$ ## Sets - $P(A) = n(A) / n$ - $A ∩ Ø = Ø$ - $A ∪ Ø = A$ - $A ∩ A' = Ø$ - $A ∪ A' = S$ - $S' = Ø$ - $Ø' = S$ ## Conditional probability - The **conditional probability of B given A** is denoted by $P(B/A)$. - $P(B/A) = P(A ∩ B)/ P(A)$ (generally) - **Mutually exclusive:** - $P(A/B) = 0$ - **Independence:** - $P(A/B) = P(B)$ - $P(A ∩ B) = P(A)P(B)$ - $P(A ∪ B) = P(A) + P(B) - P(A)P(B)$ - **In general**: $P(A ∩ B) = P(A/B)P(B) = P(B/A)P(A)$ ## Useful properties - $P(S/E) = P(E/S) = 1$ - $P((A ∪ B)/F) = P(A/F) + P(B/F) - P((A ∩ B)/F)$ - $P(E'/F) = 1 - p(E/F)$ ## Bayes' Theorem - **Total probability rule**: $P(A) = P(A/E_1)P(E_1) + P(A/E_2)P(E_2)$ (Where $E_1$ and $E_2$ are disjoint) - **Bayes' Theorem**: $P(A/B) = P(A)P(B/A) / P(B)$