Discrrete mathematics for computer science conditional warmup

Conditional Probability Warm-Up Remember how to estimate how many people answer yes to embarrassing question P.

Conditional Probability

Warm-Up

Remember how to estimate how many people answer yes

to embarrassing question P

Calculating Pr(P) from

Pr(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)

• Pr(P) = 2 ∙ Pr(H∪P) – 1

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