Probability 2

Area of a Square 100%

Area of Green Square (X) X = 25%

Area of NOT-X ~X = 75%

Formula Area(~X) = 1 – Area(X)

Area of X X = 25%

Area of A X = 25% A = 25%

Area of X v A X = 25% A = 25% 50%

Formula If X and A are non-overlapping, then Area(X v A) = Area(X) + Area(A)

Area of Y Y = 50%

Area of Z Z = 50%

Area of Y or Z Y v Z = 75%

Formula Area(Y v Z) = Area(Y) + Area(Z) – Area(Y & Z)

Area of Y & Z Y & Z = 25%

Independence Y and Z are independent: knowing that a point is in Y does not increase the probability that it’s in Z, because half of the points in Y are in Z and half are not.

Formula If Y and Z are independent, then Area(Y & Z) = Area(Y) x Area(Z)

Area of Z Z = 50%

Area of B 50%

Area of B v Z 62.5%

Area(Z & B) Z & B = 37.5%

Correlated Areas B and Z are not independent. 75% of the points in Z are also in B. If you know that a point is in Z, then it is a good guess that it’s in B too.

Formula Area(B & Z) = Area(B) x Area(Z/ B) This is the percentage of B that is in Z: 75%

Area(Z & B) Z & B = 37.5%

Area(Z & B) Z & B = 37.5% Z = 50%

Conditional Areas Area(Z/ B) = Area(Z & B) ÷ Area (B) = 37.5% ÷ 50% = 75%

Formula Area(B & Z) = Area(B) x Area(Z/ B) = 50% x 75% = 37.5%

Area(Z & B) Z & B = 37.5%

Rules Area(~P) = 1 – Area(P) Area(P v Q) = Area(P) + Area(Q) – Area(P & Q) Area(P v Q) = Area(P) + Area(Q) for non-overlapping P and Q Area(P & Q) = Area(P) x Area(Q/ P) Area(P & Q) = Area(P) x Area(Q) for independent P and Q

Rules Pr(~φ) = 1 – Pr(φ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) when φ and ψ are mutually exclusive Pr(φ & ψ) = Pr(φ) x Pr(ψ/ φ) Pr(φ & ψ) = Pr(φ) x Pr(ψ) when φ and ψ are independent

In-Class Exercises

1. Pr(P) = 1/2, Pr(Q) = 1/2, Pr(P & Q) = 1/8, what is Pr(P v Q)? 2. Pr(R) = 1/2, Pr(S) = 1/4, Pr(R v S) = 3/4, what is Pr(R & S)? 3. Pr(U) = 1/2, Pr(T) = 3/4, Pr(U & ~T) = 1/8, what is Pr(U v ~T)?

Known: Pr(P) = 1/2, Known: Pr(Q) = 1/2, Known: Pr(P & Q) = 1/8 Unknown: Pr(P v Q)

Rules Pr(~φ) = 1 – Pr(φ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) when φ and ψ are mutually exclusive Pr(φ & ψ) = Pr(φ) x Pr(ψ/ φ) Pr(φ & ψ) = Pr(φ) x Pr(ψ) when φ and ψ are independent

Pr(P v Q) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(P v Q) = Pr(P) + Pr(Q) – Pr(P & Q) = 1/2 + Pr(Q) – Pr(P & Q) = 1/2 + 1/2 – Pr(P & Q) = 1/2 + 1/2 – 1/8 = 7/8 Known: Pr(P) = 1/2, Known: Pr(Q) = 1/2, Known: Pr(P & Q) = 1/8

Known: Pr(R) = 1/2 Known: Pr(S) = 1/4 Known: Pr(R v S) = 3/4 Unknown: Pr(R & S)?

Not Helpful: More than One Unknown Pr(~φ) = 1 – Pr(φ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) when φ and ψ are mutually exclusive Pr(φ & ψ) = Pr(φ) x Pr(ψ/ φ) Pr(φ & ψ) = Pr(φ) x Pr(ψ) when φ and ψ are independent

This Is What You Want Pr(~φ) = 1 – Pr(φ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) when φ and ψ are mutually exclusive Pr(φ & ψ) = Pr(φ) x Pr(ψ/ φ) Pr(φ & ψ) = Pr(φ) x Pr(ψ) when φ and ψ are independent

Pr(R & S) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(R v S) = Pr(R) + Pr(S) – Pr(R & S) 3/4 = Pr(R) + Pr(S) – Pr(R & S) 3/4 = 1/2 + Pr(S) – Pr(R & S) 3/4 = 1/2 + 1/4 – Pr(R & S) 3/4 = 3/4 – Pr(R & S) Known: Pr(R) = 1/2 Known: Pr(S) = 1/4 Known: Pr(R v S) = 3/4

Known: Pr(U) = 1/2 Known: Pr(T) = 3/4 Known: Pr(U & ~T) = 1/8 Unknown: Pr(U v ~T)?

Rules Pr(~φ) = 1 – Pr(φ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) when φ and ψ are mutually exclusive Pr(φ & ψ) = Pr(φ) x Pr(ψ/ φ) Pr(φ & ψ) = Pr(φ) x Pr(ψ) when φ and ψ are independent

Pr(~T) Pr(~φ) = 1 – Pr(φ) Pr(~T) = 1 – Pr(T) = 1 – 3/4 = 1/4 Known: Pr(U) = 1/2 Known: Pr(T) = 3/4 Known: Pr(U & ~T) = 1/8 Known: Pr(~T) = 1/4

Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Known: Pr(U) = 1/2 Known: Pr(T) = 3/4 Known: Pr(U & ~T) = 1/8 Known: Pr(~T) = 1/4

Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Known: Pr(U) = 1/2 Known: Pr(T) = 3/4 Known: Pr(U & ~T) = 1/8 Known: Pr(~T) = 1/4

Pr(U v ψ) = Pr(U) + Pr(ψ) – Pr(U & ψ) Known: Pr(U) = 1/2 Known: Pr(T) = 3/4 Known: Pr(U & ~T) = 1/8 Known: Pr(~T) = 1/4

Pr(U v ψ) = Pr(U) + Pr(ψ) – Pr(U & ψ) Known: Pr(U) = 1/2 Known: Pr(T) = 3/4 Known: Pr(U & ~T) = 1/8 Known: Pr(~T) = 1/4

Pr(U v ~T) = Pr(U) + Pr(~T) – Pr(U & ~T) Known: Pr(U) = 1/2 Known: Pr(T) = 3/4 Known: Pr(U & ~T) = 1/8 Known: Pr(~T) = 1/4

Pr(U v ~T) Pr(U v ~T) = Pr(U) + Pr(~T) – Pr(U & ~T) = 1/2 + Pr(~T) – Pr(U & ~T) = 1/2 + 1/4 – Pr(U & ~T) = 1/2 + 1/4 – 1/8 = 5/8 Known: Pr(U) = 1/2 Known: Pr(T) = 3/4 Known: Pr(U & ~T) = 1/8 Known: Pr(~T) = 1/4

More Exercises 4. Suppose I flip a fair coin three times in a row. What is the probability that it lands heads all three times? 5. Suppose I flip a fair coin four times in a row. What is the probability that it does not land heads on any of the flips?

Problem #4 4. Suppose I flip a fair coin three times in a row. What is the probability that it lands heads all three times? Known: Pr(F) = 1/2 Known: Pr(S) = 1/2 Known: Pr(T) = 1/2 Unknown: Pr((F & S) & T)

Rules Pr(~φ) = 1 – Pr(φ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) when φ and ψ are mutually exclusive Pr(φ & ψ) = Pr(φ) x Pr(ψ/ φ) Pr(φ & ψ) = Pr(φ) x Pr(ψ) when φ and ψ are independent

Pr((F & S) & T) Pr(φ & ψ) = Pr(φ) x Pr(ψ) Pr((F & S) & T) = Pr(F & S) x Pr(T) = Pr(F) x Pr(S) x Pr(T) = 1/2 x 1/2 x 1/2 = 1/8

Problem #5 5. Suppose I flip a fair coin four times in a row. What is the probability that it does not land heads on any of the flips? Known: Pr(F) = 1/2 Known: Pr(S) = 1/2 Known: Pr(T) = 1/2 Known: Pr(L) = 1/2 Unknown: Pr((~F & ~S) & (~T & ~L))

Rules Pr(~φ) = 1 – Pr(φ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) when φ and ψ are mutually exclusive Pr(φ & ψ) = Pr(φ) x Pr(ψ/ φ) Pr(φ & ψ) = Pr(φ) x Pr(ψ) when φ and ψ are independent

Using ~ Rule Pr(~F) = 1 – 1/2 = 1/2 Pr(~S) = 1 – 1/2 = 1/2 Pr(~T) = 1 – 1/2 = 1/2 Pr(~L) = 1 – 1/2 = 1/2

Pr((~F & ~S) & (~T & ~F)) Pr(φ & ψ) = Pr(φ) x Pr(ψ) Pr((~F & ~S) & (~T & ~F)) = Pr(~F & ~S) x Pr(~T & ~F) = Pr(~F) x Pr(~S) x Pr(~T & ~F) = Pr(~F) x Pr(~S) x Pr(~T) x Pr(~F) = 1/2 x 1/2 x 1/2 x 1/2 = 1/16

Conditional Probability

Area of P P = 50%

Area of Q Q = 50%

Area(P & Q) P & Q = 37.5%

Area(P & Q) out of Area(Q) P & Q = 75%

Conditional Areas Area(P/ Q) means: the percentage of (P & Q) points out of all Q-points. Area(P/ Q) = Area(P & Q) Area(Q)

Conditional Probabilities Pr(P/ Q) means: the percentage of (P & Q) possibilities out of all Q- possibilities. Pr(P/ Q) = Pr(P & Q) Pr(Q)

Probabilistic Generalizations Our probabilistic generalizations usually express conditional probabilities: 90% of bankers are rich ≠ the probability of someone being rich is 90% ≠ the probability of someone being a banker is 90% ≠ the probability of someone being a rich banker is 90% = the probability of someone being rich assuming that they are a banker is 90%

Coin Flips Suppose I flip a coin twice. The probability that it will land heads on the first flip is 50%. The probability that it will land heads on the second flip is 50%.

Coin Flips FirstSecond Heads Tails Heads Tails

Coin Flips Assuming nothing, what is the probability that both coins land heads? Pr(F & S) = ?

Pr(F & S) = 1/4 FirstSecond Heads Tails Heads Tails

How Did We Calculate That? Since two coin flips are independent, we know: Pr(F v S) = Pr(F) x Pr(S) = 50% x 50% = 25%

Coin Flips Assuming that one of the coin flips lands heads, what is the probability that the other one also lands heads? Pr(F & S/ F v S) = ?

Ignore the Possibilities with No Heads FirstSecond Heads Tails Heads Tails

Pr(F & S/ F v S) = 1/3 FirstSecond Heads Tails Heads

How Did We Calculate That? Pr(F & S/ F v S) = = = = 1/3 Pr((F & S) & (F v S)) Pr(F v S) Pr(F& S) Pr(F v S) 25% 75%

Coin Flips Assuming that the first coin flip lands heads, what is the probability that the other one also lands heads? Pr(F & S/ F) = ?

Ignore Possibilities Where First Is Not Heads FirstSecond Heads Tails Heads Tails

Pr(F & S/ F) = 50% FirstSecond Heads Tails

How Did We Calculate That? Pr(F & S/ F) = = = = 1/2 Pr((F & S) & F) Pr(F) Pr(F& S) Pr(F) 25% 50%

Bayes’ Theorem

Simple Algebra Pr(B/ A) = Pr(B & A) ÷ Pr(A) Pr(A) x Pr(B/ A) = [Pr(A) x Pr(B & A)] ÷ Pr(A) Pr(A) x Pr(B/ A) = Pr(B & A) Pr(A & B) = Pr(B/ A) x Pr(A)

Bayes’ Theorem Pr(A/ B) = Pr(A & B) ÷ Pr(B) Pr(A/ B) = [Pr(B/ A) x Pr(A)] ÷ Pr(B)

Bayes’ Theorem Baye’s theorem lets us calculate the probability of A conditional on B when we have the probability of B conditional on A.

Base Rate Fallacy There are ½ million people in Russia are affected by HIV/ AIDS. There are 150 million people in Russia.

Base Rate Fallacy Imagine that the government decides this is bad and that they should test everyone for HIV/ AIDS.

The Test If someone has HIV/ AIDS, then : 95% of the time the test will be positive (correct) 5% of the time will it be negative (incorrect)

The Test If someone does not have HIV/ AIDS, then: 95% of the time the test will be negative (correct) 5% of the time will it be positive (incorrect)

Suppose you test positive. We’re interested in the conditional probability: what is the probability you have HIV assuming that you test positive. We’re interested in Pr(HIV = yes/ test = pos)

Known: Pr(sick) = 1/300 Known: Pr(positive/ sick) = 95% Known: Pr(positive/ not-sick) = 5% Unknown: Pr(positive) Unknown: Pr(sick/ positive)

Pr(positive) = True positives + false positives = [Pr(positive/ sick) x Pr(sick)] + [Pr(positive/ not-sick) x Pr(not-sick)] = [95% x 1/300] + [5% x 299/300] = 5.3% Known: Pr(sick) = 1/300 Known: Pr(positive/ sick) = 95% Known: Pr(positive/ not-sick) = 5%

Known: Pr(sick) = 1/300 Known: Pr(positive/ sick) = 95% Known: Pr(positive) = 5.3% Unknown: Pr(sick/ positive)

Pr(sick/ positive) Pr(A/ B) = [Pr(B/ A) x Pr(A)] ÷ Pr(B) Pr(sick/ positive) = [Pr(positive/ sick) x Pr(sick)] ÷ Pr(positive) = [95% x Pr(sick)] ÷ Pr(positive) = [95% x 1/300] ÷ Pr(positive) = [95% x 1/300] ÷ 5.3% = 5.975% Known: Pr(sick) = 1/300 Known: Pr(positive/ sick) = 95% Known: Pr(positive) = 5.3%