Chapter 2.2 - Math3200 Intermediate Probability and Statistics Prof Nan Lin Department of Mathematics Washington University Agenda Conditional

# Chapter 2.2 - Math3200 Intermediate Probability and...

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Math3200 Intermediate Probability and Statistics Prof. Nan Lin Department of Mathematics Washington University

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Agenda Conditional Probability Independence
Conditional Probability Assume 𝑃 ? > 0 , the conditional probability of B given A, 𝑃(?|?) , is 𝑃 ? ? = 𝑃 ? ∩ ? 𝑃 ? = 𝑃 ?&? 𝑃 ? Validity: Since ? ∩ ? ⊆ ? , 𝑃(? ∩ ?) ≤ 𝑃(?) , hence, 0 ≤ 𝑃 ? ? ≤ 1 . Similarly, 𝑃 ? 𝑐 ? = 𝑃 ? 𝑐 &? 𝑃 ?

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Special cases
Example: roll two dice Two fair dice are rolled. What is the (conditional) probability that the sum of the two faces is 6 given that the two dice are showing different faces? Sample space: 𝑆 = { ?, ? : 1 ≤ ? ≤ 6,1 ≤ ? ≤ 6} ? = { ?, ? : (?, ?) ∈ 𝑆, ? ≠ ?} , 𝑃 ? = 30 36 ? = { 1,5 , 2,4 , 3,3 , 4,2 , (5,1)} , 𝑃 ? = 5 36 ? ∩ ? = 1,5 , 2,4 , 4,2 , 5,1 , 𝑃 ? ∩ ? = 4 36 𝑃 ? ? = 𝑃 ?∩? 𝑃 ? = 4 30 = 2 15 < 𝑃(?)

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Example: roll two dice (cont) Two fair dice are rolled. What is the (conditional) probability that the sum of the two faces is 7 given that the two dice are showing different faces? Sample space: 𝑆 = { ?, ? : 1 ≤ ? ≤ 6,1 ≤ ? ≤ 6} ? = { ?, ? : (?, ?) ∈ 𝑆, ? ≠ ?} , 𝑃 ? = 30 36 C = { 1,6 , 2,5 , 3,4 , 4,3 , 5,2 , (6,1)} , 𝑃 ? = 6 36 C ∩ ? = ?, 𝑃 ? ∩ ? = 6 36 𝑃 ? ? = 𝑃 ?∩? 𝑃 ? = 6 30 = 1 5 > 𝑃(?)
Example: roll two dice (cont) Two fair dice are rolled. What is the (conditional) probability that the first die shows a 6 given that the two dice are showing different faces?

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