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) = .5
• 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 + .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 = 2 – 1/r

If r = ¾, Pr(P|R) = 2 – 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