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Review: Conditional Probability

1/23/2020

Sofya Raskhodnikova; Randomness in Computing

Conditional Probability

The conditional probability of event 𝐹 given event 𝐺 is Pr 𝐹 𝐺 = Pr(E ∩ 𝐺) Pr(𝐺) ⋅

Well defined only if Pr 𝑮 ≠ 𝟏

Review question

Card dealing We deal two cards. What is the probability that the second card is an ace, given that the first is an ace?

• A. 3/52
• B. 3/51
• C. 4/52
• D. 5/52
• E. None of the answers above are correct.

1/23/2020

Sofya Raskhodnikova; Randomness in Computing

Product rule

1/23/2020

Sofya Raskhodnikova; Randomness in Computing

For any two events 𝐹1 and 𝐹2,

Pr 𝐹1 ∩ 𝐹2 = Pr(𝐹1) ⋅ Pr 𝐹2|𝐹1 .

For all events 𝐹1, … , 𝐹𝑜,

Pr(∩𝑗=1

𝑜

𝐹𝑗) = Pr 𝐹1 ⋅ Pr 𝐹2 𝐹1 ⋅ … ⋅ (𝐹𝑜| ∩𝑗=1

𝑜−1 𝐹𝑗)

Law of Total Probability

1/23/2020

Sofya Raskhodnikova; Randomness in Computing

For any two events 𝐵 and 𝐹,

Pr 𝐵 = Pr(𝐵 ∩ 𝐹) + Pr(𝐵 ∩ 𝐹) = Pr(𝐵|𝐹) ⋅ Pr 𝐹 + Pr(𝐵|𝐹) ⋅ Pr 𝐹 Let A be an event and let 𝐹1, … , 𝐹𝑜 be mutually disjoint events whose union is Ω. Pr(𝐵) =

𝑗∈ 𝑜

Pr(𝐵 ∩ 𝐹𝑗) =

𝑗∈ 𝑜

Pr(𝐵 ∣ 𝐹𝑗) ⋅ Pr(𝐹𝑗) .

Bayes’ Law

1/23/2020

Sofya Raskhodnikova; Randomness in Computing

For any two events 𝐵 and 𝐹 with Pr(A) ≠ 0,

Pr 𝐹|𝐵 = Pr(𝐵|𝐹) ⋅ Pr(𝐹) Pr(𝐵) Let A be an event with Pr(A) ≠ 0 and let 𝐹1, … , 𝐹𝑜 be mutually disjoint events whose union is Ω. Pr(𝐹

𝑘|𝐵) = Pr(𝐹 𝑘 ∩ 𝐵)

Pr(𝐵) = Pr(𝐵|𝐹

𝑘) ⋅ Pr(𝐹 𝑘)

𝑗∈ 𝑜 Pr(𝐵 ∣ 𝐹𝑗) ⋅ Pr(𝐹𝑗)