# 05 01 01 065 04 025 m chen iecuhk engg2430c

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Unformatted text preview: -pick IE graduate gets a monthly salary at least 20K HKD ? 0.5 × 0.1 + 0.1 × 0.65 + 0.4 × 0.25 M. Chen ([email protected]) ENGG2430C lecture 4 5 / 19 Total Probability Law: Divide and Conquer (D. P. Bertsekas & J. N. Tsitsiklis, Introduction to Probability, Athena Scientiﬁc Publishers, 2002) Noticing B = B ∩ Ω = [A1 ∩ B ] ∪ [A2 ∩ B ] ∪ [A3 ∩ B ] and sets Ai ∩ B (i = 1, 2, 3) are disjoint, we get P (B ) = P (A1 ∩ B ) + P (A2 ∩ B ) + P (A3 ∩ B ) = P (A1 )P (B |A1 ) + P (A2 )P (B |A2 ) + P (A3 )P (B |A3 ) M. Chen ([email protected]) ENGG2430C lecture 4 6 / 19 Binary Symmetric Channel (BSC) * Error probability ε : P (0r |1t ) = P (1r |0t ) = ε . Question: Assume P (0t ) = P (1t ) = 0.5, P (1r ) =? If I know what is transmitted, then it is easy: P (1r |1t ) = 1 − ε , P (1r |0t ) = ε . Without such knowledge, it is a weighted sum of these two conditional probabilities: P (1r ) = P (1r |1t )P (1t ) + P (1r |0t )P (0t ) = (1 − ε )0.5 + ε · 0.5 = 0.5. The answer does not depend on ε . Intuitive explanation? M. Chen ([email protected]) ENGG2430C lecture 4 7 / 19 Bayes’ Rule Law for combining evidence. Infer from observations. (D. P. Bertsekas & J. N. Tsitsiklis, Introduction to Probability, Athena Scientiﬁc Publishers, 2002) “Prior” probability: P (Ai ), i = 1, 2, 3 For each i , we know P (B |Ai ) Compute P (A1 |B ) P (A1 |B ) = (Apply total probability theorem on P (B )) = M. Chen ([email protected]) ENGG2430C lecture 4 P (B |A1 )P (A1 ) P (A1 ∩ B ) = P (B ) P (B ) P (B |A1 )P (A1 ) ∑j P (Aj )P (B |Aj ) 8 / 19 Binary Symmetric Channel (BSC) * Error probability ε : P (0r |1t ) = P (1r |0t ) = ε . Question: Assume P (0t ) = P (1t ) = 0.5. Given that we receive 1, what is probability that 1 is transmitted? P (1t |1r ) = M. Chen ([email protected]) P (1t , 1r ) P (1r |1t )P (1t ) = = 1 − ε. P (1r ) P (1r ) ENGG2430C lecture 4 9 / 19 Example: False-Positive Puzzle * P (having the disease) = 0.001, P (healthy) = 0.999. Question: Given that the person tested positive,...
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