Notes - Matwa - PAS 2025
P and C
(P_k^n)=n!/(n-k)!
n permute k. Permutation of elements with replacement/repetition. The formula returns all possible samplings of length k from vector of length n.
(C_k^n)=([n],[k])=n!/(n!⋅(n-k)!)
n choose k. The number of ways to choose k items from a set of n distinct items without regard to the order in which they are chosen.
Binomial
(x+y)n=(∑_k=0^n)(([n],[k])⋅x(n-k)⋅yk)
Events
P(A)=Favorable/Possible
The formula for probabilities is just the number of favorable outcomes divided by the number of possible outcomes.
P((A^∁))=1-P(A)
The probability where A does not happen is called the complement.
Addition Rule
Use addition when you want A OR B.
P(A∪B)=P(A)+P(B)
The above is the formula if A and B are mutually exclusive. Subtract by P(A∩B) if they aren't mutually exclusive, aka they can happen together.
You should subtract by P(A∩B) due to the Inclusion Exclusion rule as to not double count occasions where both A and B happen.
Multiplication Rule
Use multiplication when we want A AND B to happen together, either at the same time or in sequence.
P(A∩B)=P(A)⋅P(B)
Use the formula above if A does not affect B, such as when throwing a dice and a coin, or throwing 2 dice at once.
P(A∩B)=P(A)⋅P((B|*A)
Use the formula above if the probability changes after something happens. P((B|*A) is the probability of B given A already happened.