Discrrete mathematics for computer science conditional warmup

Conditional Probability Warm-Up Remember how to estimate how many people answer yes to embarrassing question P Calculating Pr(P) from Pr(H∪P) H P • Because H and P are independent events, Pr(H∩P) = Pr(H) ∙ Pr(P) • Pr(H) = • Pr(H∪P) = Pr(H) + Pr(P) - Pr(H∩P) • So Pr(P) = Pr(H∪P) - Pr(H) + Pr(H)∙Pr(P) • Pr(P) = Pr(H∪P) -.5 + ∙ Pr(P) Now Suppose You Raise Your Hand: How Suspicious Should I Be of You? • That is, what is Pr(P | H∪P)? • Let R = H∪P, r = Pr(R), p = Pr(P) = 2r-1 • We want Pr(P | R) = Pr(P∩R)/Pr(R) • But P∩R = P∩(H∪P) = P • So Pr(P∩R)/Pr(R) = Pr(P)/Pr(R) = p/r = (2r-1)/r = – 1/r If r = ¾, Pr(P|R) = – 4/3 = 2/3 Important Lessons! • So if r = 5, Pr(P|R) = 0; if r = 1, Pr(P|R) =1 • With only a finite sample, impossible to calculate probabilities precisely FINIS