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Topic B4 — Introducing Probability

Table of contents
  1. Topic B4 — Introducing Probability
    1. Permutations vs. Combinations
    2. Sample space vs. Event
    3. Union, Intersection and Complement
    4. Unconditional vs. Conditional probability
    5. Dependent vs. Independent events
    6. Multiplication
    7. Total probability rule
    8. Bayes’ Theorem

Permutations vs. Combinations

Permutations Order Matters

  • You have spots to fill (ChoiceSize)
  • Draw names from a box (SizePool)
  • Names are unique, and you don’t put the names back in the box
  • Formula: $\frac{\mathit{SizePool}!}{\mathit{(SizePool-ChoiceSize)}!} = \mathit{SizePool} \times (\mathit{SizePool}-1) \times \dots \times (\mathit{SizePool}-\mathit{ChoiceSize}) $
  • Simplify by canceling out factorials

Combinations Order Doesn’t Matter

  • Intuition: bigger or smaller than permutations? Smaller, because less ways to do it.
  • In how many ways can a list have the same elements? $ \mathit{ChoiceSize}! $ These are the number of similar lists with different orders!
  • Divide permutations by that number

Example

  • Pool size 4
  • Choice size 2

Sample space vs. Event

Kind of like population and sample

  • Sample space: all the possible outcomes
  • Events: a sub set of the sample space

Dice example

  • $S={1,2,3,4,5,6}$
  • $A=2,3,4$
  • $B=4,5,1$
  • $A^c = 1,5,6$

Union, Intersection and Complement

Union:

  • Add the two set, don’t repeat the common events
  • One or the other occurs
  • $P(A \cup B)$

Intersection

  • Common between events
  • Both occur
  • $P(A \cap B)$

Complements

  • “The contrary”
  • Does not occur

Unconditional vs. Conditional probability

Unconditional = “Normal”

Conditional on something happening: the prob of the intersection between events over the probability of the condition

$P(A\mid B) = \frac{P(A \cap B)}{P(B)}$

“Occur if the other occurs”

Dependent vs. Independent events

Independent

  • If the conditional prob = the uncoditional (“regular”) one
  • $P(A\mid B)=P(A)$
  • Occurs with the same probability whether or not the other occurs

Dependent

  • $P(A\mid B) \neq P(A)$

Multiplication

$ P(A \cap B) = P(A \mid B) \times P(B) = P(B \mid A) \times P(A)$

Both occur = Prob that A occurs when B occurs, times prob of B occurring

Total probability rule

An event conditional on two, mutually exclusive, collectively exhaustive events. –> Venn space divided in 2 by $B$ and $B^c$, and $A$ conditional on both

$P(A) = P(A\cap B) + P(A\cap B^c) $

Substituting with the formula above, we get

$P(A) = P(A \mid B) \times P(B) + P(A \mid B^c) \times P(B^c) $

Bayes’ Theorem

A way to update probabilities: from prior $P(B)$ to updated (i.e. conditional on something happening) $P(B\mid A)$

$ P(B\mid A) = \frac{P(A \mid B) \times P(B)}{P(A \mid B) \times P(B) + P(A \mid B^c) \times P(B^c)} = \frac{P(A \cap B)}{P(A)}$

Supplementary Bayes’ theorem material

Great video explanation (and great YouTube channel)