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Summary STA1000 Modules 1 and 2

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A comprehensive summary of modules 1 and 2 for STA1000 including additional notes.

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  • September 16, 2019
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  • 2019/2020
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Module 1: Probability


Work Unit 1: Introducing Probability

No . of ways of getting selected outcome
Pr (an outcome)=
Total number of possible outcomes

Payout for winning number
Win %= ×100
Amount bet across all numbers

House Advantage =100−Win %

payout
Fair payout for a win=
win %

expected profit∨loss
% return on amount bet=
bet amount

House Advantage =−% return

Roulette uses the numbers 0–37, but the zero is ‘dead’. There are actually 37 options, but when
using probabilities there are 36 options.

Work Unit 2: Set Theory, Probability Axioms and Theorems

Sample Space – S
- Universal set; all objects under consideration

Subset – ⊂ or ⊃
- If one set is contained in another

Empty Set – Ø or {}
- Has no members
- If the intersection of two sets is an empty set, then the sets are disjoint/mutually exclusive

Intersection (and) – ∩
- The intersection of any two sets is the set that contains precisely those elements which belong
to both sets
- Multiply

Union (or) – ∪
- The union of A and B is the set that contains the elements that belong to either A or B
- Add

, Complement – Ᾱ
- Given the sample space S, the complement of set A is the set of elements in S which are not in A



Venn Diagrams




Exhaustive – uses up all of the sample space
Pairwise Mutually Exclusive – three or more events are said to be pairwise mutually exclusive if no
two of them are compatible

Kolmogorov’s Axioms of Probability:
1. 0 ≤ Pr(Event) ≤ 1
2. Pr(S) = 1
3. If Pr(A∩B) = Ø then Pr(A∪B) = Pr(A) + Pr(B)

Relative Frequency
- the number of times that an event (r) occurs during (n) trials. As the number of trials increase, so
the relative frequency tends to get nearer the probability of selected event (r).


Relative Frequency = ( Number of ×event occurs
Number of trials ) r
∨( )
n

Theorems:
1. Pr(Ᾱ) = 1 – Pr(A); if A is a subset of S
2. Pr(B) = Pr(A∩B) + Pr(Ᾱ∩B); if A ⊂ S & B ⊂ S

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