Lecture 15 - Law of Total Probability

LECTURE 15 Law of total probability P(E) = P(E|H₁) P(E|U₁) P(E|U₁ U₂) P(E|U₁ U₂ U₃) P(E|U₁ U₂ ... Uₖ) P(E) = ΣP(Hᵢ)P(E|Hᵢ) Assuming that • the basic outcome (Ω) have the same probability to occur • the four events Hᵢ, i=1,...,k are mutually exclusive and collectively exhaustive Compute P(E) using the law of total probability: P(E) = ΣP(Hᵢ)P(E|Hᵢ) = P(H₁)P(E|H₁) + P(H₂)P(E|H₂) + P(H₃)P(E|H₃) + P(H₄)P(E|H₄) P(E|H₁) = 1/4 P(E|H₂) = 1/7, P(E|H₃) = 1/4, P(E|H₄) = 2/5 = 1/4 * 0 + 7/20 * 1/7 + 4/20 * 1/4 + 5/20 * 2/5 = 1/20 + 1/20 + 1/10 = 4/20 = 4/5 = 20% Example: smallpox virus in Boston 1721 The data concern 6224 people exposed to the smallpox virus in Boston in 1721. Of these, 244 were exposed on a voluntary basis to the disease in a controlled manner by doctors. It is then known who survived the epidemic and who did not. Exposed Yes No Total Survived Yes 238 5130 5374 2 variables: YES, NO No 6 844 950 Total 244 5980 6224 S survived, Exexposed a. For a subject selected at random, what is the probability to survive the epidemic? P(S) = #survived / #total = 5374 / 6224 = 0.86 = 86% b. For a subject selected at random, what is the probability to be exposed? P(E) = #exposed / #total = 244 / 6224 = 0.039 = 3.9% c. For a random individual, what is the probability that she was exposed and survived? P(S|E) = #exposed and survived / #exposed = 238 / 244 = 0.98 = 98% d. For an exposed subject selected at random, what is the probability to survive? P(S|E) = #survived / #exposed = 238 / 244 = 0.974 = 97.4% e. For a not exposed subject selected at random, what is the probability to survive? P(S|E') = #survived / #not exposed = 5136 / 5980 = 0.859 = 85.9% f. Verify that you obtain the same result in point a. of P(S) using the law of total probability P(S) = P(S|E)P(E) + P(S|E')P(E') = 238/6224 * 244/6224 + 5136/6224 * 5980/6224 = 0.86 = 86% Example: US president elections 2020 The table below shows the results of the popular vote in the last U.S. presidential elections reported by the Federal Election Commission. Candidates are listed individually if they received more than 0.1% of the popular vote. Americans eligible to vote in 2020 were 239,661,805 Presidential candidate Party Popular vote Joseph Robinette Biden Jr. Democratic Republican 81,268,924 Donald John Trump 74,216,154 Jo Jorgensen Libertarian 1,865,724 Howard Gresham Hawkins Green 405,035What would happen if we ran all the 100,000 tests? - The 99,999 innocents all have incompatible DNA, but due to FP, on average 0.1% is positive (10) Other 627,566 Total 158,383,403 d. What is the probability that an eligible voter selected at random actually voted? P(vote) = 158,383,403 / 239,661,805 = 0.661 = 66% b. What is the probability that an eligible voter selected at random voted for the Democratic party? P(Democratic) = 81,268,924 / 239,661,805 = 0.339 = 33% c. What is the probability that an eligible voter selected at random, went to vote and voted Democratic? P(Vote Democratic) = P(Democratic) = 0.339 = 33% d. Knowing that a voter selected at random went to vote, what is the probability that she voted Democratic? P(Democratic|vote) = P(vote Democratic) / P(vote) = 0.339 / 0.661 = 0.513 = 51%