Scientific Question: Question created by an experimenter

Experiment: Task to collect info that answers a scientific question

Sample Space: $\Omega$ set of all possible outcomes of an experiment

• E.g. $\Omega=\{H,T\}$ is the sample space for flipping a coin
• E.g. $\Omega=\{HH,TT,HT,TH\}$ is the sample space for flipping two coins

Events: Subsets of $\Omega$

• May be combined by set operations $\cup,\cap,A^c$
• $1 \ge P(A) \ge 0$: Probability of A
• $P(\Omega)=1=P(A)+P(A^c)$

Set Operations:

• Intersection: $\cap$, Union: $\cup$
• $A=A \cap B^C+A \cap B$
• $A\cup B = (A \cap B^C) \cup (A\cap B) \cup (A^C \cap B)$
• $P(A\cup B)=P(A)+P(B)-P(A \cap B)$

Properties

• Mutually Exclusive / Disjoint: No common elements between the two sets $A\cap B=\emptyset$
  • $P(A \cup B) = P(A) + P(B)$ if $A,B$ are mutex
• DeMorgan’s Laws: $(A \cup B)^c = A^c \cap B^c, (A\cap B)^c = A^c \cup B^c$
• Independent: $A$ and $B$ are independent if $P(A\cap B) = P(A) P(B)$ E.g. two coin flips

Permutation: Ordered arrangement of $n$ distinct objects (order matters)

• $P^n_r=n(n-1)(n-2)\dots(n-r+1)=\frac{n!}{(n-r)!}$