STA256 Lecture 03

• STA256H5F_Fall2025_Ch1_Probability_Distributions.pdf
• If we don't have disjoint probabilities, then we have the Addition Law of Probability
• For a single event, Property of Complements
• Strategy to solve probability questions:
  • Before reading the question, write down and visualize all the information.
    • Drawing these visuals are good, this makes it very easy to see how these sets and axioms work.
• Example 1:
  • $P(A)=0.01$
  • $P(B)=0.02$
  • $P(A\cap B)=0.40$
  • $P(A\cup B)=0.43$
  • $P((A\cup B{)}^c)=0.57$
  • Find the probability that protein is referenced by either:
    • That's $P(A\cup B)=0.43$
    • Additive Property:
      • $P(A\cup B)=P(A)+P(B)-P(A\cap B)$
      • $P(A\cup B)=0.41+0.42-0.40$
      • $P(A\cup B)=0.43$
    • $43\mathrm{\%}$
  • Find the probability that protein is not referenced by either:
    • $P(A\cup B)=1-P((A\cup B{)}^c)$
    • $0.43=1-P((A\cup B{)}^c)$
    • $0.43-1=-P((A\cup B{)}^c)$
    • $1-0.43=P((A\cup B{)}^c)$
    • $0.57=P((A\cup B{)}^c)$
• Cardinality
• Crossing:
  • $A=\{a_1,a_2,\dots ,a_m\}$
  • $B=\{b_1,b_2,\dots ,b_n\}$
  • See the number of pairs:
    • $\begin{array}{c}\{\{a_1,b_1\},\{a_1,b_2\}\dots ,\{a_1,b_n\}\\ \{a_2,b_1\},\{a_2,b_2\},\dots ,\{a_2,b_n\}\\ \{a_m,b_1\},\{a_m,b_2\}\dots ,\{a_m,b_n\}\}\end{array}$
    • We have $n$ columns and $m$ rows.
    • That's the pairs of $m\times n$
• Factorial
• Permutations
• Combinations
• Inclusion-Exclusion Formula
• Continuity Theorem of Probability
• Boole's Theorem
• Conditional Probability and Independence
• Multiplication Law of Probability
• Statistical Independence