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Total Probability Theorem
- Claim. If B ⊂ A then Pr(B) ≤ Pr(A).
- Proof. A = B ∪ (A \ B), so
Pr(A) = Pr(B) + Pr(A \ B) ≥ Pr(B).
- Def. The events A 1 ,... , An form a partition of the sample space Ω if 1. Ai are mutually exclusive: Ai ∩ Aj = ∅ for i 6 = j. 2. A 1 ∪... ∪ An = Ω.
- Total Probability Theorem. Let A 1 ,... , An be a partition of Ω. For any event B,
Pr(B) =
∑^ n
j=
Pr(Aj) Pr(B|Aj).
- Proof. B = ∪(B ∩ Aj) (disjoint union), so
Pr(B) =
∑^ n
j=
Pr(B ∩ Aj).
The theorem follows from Pr(B∩Aj) = Pr(Aj) Pr(B|Aj).
- The latter holds for Aj with Pr(Aj) = 0 if we define Pr(Aj) Pr(B|Aj) := 0 since then P (B ∩ Aj) = 0
Example
· 60% of registered voters are Republicans · 30% are Democrats · 10% are Independents.
- When those voters were asked about increasing mili- tary spending · 40% of Republicans opposed it · 65% of the Democrats opposed it · 55% of the Independents opposed it.
- What is the probability that a randomly selected voter in this county opposes increased military spending?
Bayes’ Theorem
- Bayes Theorem. Let A 1 ,... , An be a partition of Ω. For any event B
Pr(Ai|B) =
Pr(Ai) Pr(B|Ai) Σnj=1 Pr(Aj) Pr(B|Aj)
Pr(Ai|B) =
Pr(Ai ∩ B) Pr(B)
=
Pr(Ai) Pr(B|Ai) Σnj=1 Pr(Aj) Pr(B|Aj)
· A registered voter from our county writes a letter to the local paper, arguing against increased mil- itary spending. What is the probability that this voter is a Democrat? · Presumably that is Pr(D|B), so by Bayes’ theo- rem:
Pr(D|B) =
AIDS
- Just for the heck of it Bob decides to take a test for AIDS and it comes back positive.
- The test is 99% effective (1% FP and FN).
- Suppose 0.3% of the population in Bob’s “bracket” has AIDS.
- What is the probability that he has AIDS?
- · Ω = {all the people in Bob’s bracket}.
· A 1 = {people in Ω with AIDS}, Pr(A 1 ) = 0. 003 · A 2 = {people in Ω without AIDS}, Pr(A 2 ) = 0. 997 · B = {people in Ω who would test positive} · Pr(B|A 1 ) = .99 and P r(B|A 2 ) =. 01 · By Bayes’ rule
Pr(A 1 |B) =
- Example. What is the probability that a randomly chosen person in the class will weigh more than 160 lbs.? · A natural random variable in this case is the weight of the selected student.
- What is the difference between the range of values our two random variables can attain?
Probability Distributions
- There is a natural probabilistic structure induced on a random variable X defined on Ω: · The set {ω ∈ Ω : X(ω) = c} is an event. · So we can ask for Pr(X = c) = Pr({ω ∈ Ω : X(ω) = c}). Example. A biased coin (Pr(H) = 2/3) is flipped twice. · Let X count the number of heads: Pr(X = 0) = Pr({T T }) = (1/3)^2 = 1/ 9. Pr(X = 1) = Pr({HT, T H}) = 2 · 1 / 3 · 2 /3 = 4/ 9. Pr(X = 2) = Pr({HH}) = (2/3)^2 = 4/ 9.
- Similarly we might be interested in:
Pr(X ≤ c) = Pr({ω ∈ Ω : X(ω) ≤ c}, and more generally, for any T ⊂ R: Pr(X ∈ T ) = Pr({ω ∈ Ω : X(ω) ∈ T }.
Pr(X ≤ 1) = Pr({T T, HT, T H}) = 1/9+4/9 = 5/ 9.
