Chen iecuhk engg2430c lecture 2 14 23 properties of

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Unformatted text preview: 3) P (A ∪ B ) = P (A) + P (B ) − P (A ∩ B ) ≤ P (A) + P (B ) P ((A ∩ B c ) ∪ (Ac ∩ B )) = P (A) + P (B ) − 2P (A ∩ B ) M. Chen (IE@CUHK) ENGG2430C lecture 2 14 / 23 Properties of Probability Law (from the textbook) M. Chen (IE@CUHK) ENGG2430C lecture 2 15 / 23 Examples Problem 1.5 from textbook: Out of the students in a class, 60% are geniuses, 70% love chocolate, and 40% fall into both categories. Determine the probability that a student, that is selected uniformly at random, is neither a genius nor a chocolate lover. (Sketch: P (A) = 0.6, P (B ) = 0.7, and P (A ∩ B ) = 0.4. P (Ac ∩ B c ) = 1 − P (A ∪ B ) = 1 − P (A) − P (B ) + P (A ∩ B ).) Problem 1.7 from textbook: A four-sided die is rolled repeatedly, until the first time (if ever) that an even number is obtained. What is the sample space for this experiment? M. Chen (IE@CUHK) ENGG2430C lecture 2 16 / 23 Sample space with an Infinite Number of Outcomes Sample space: {1, 2, . . .} Probability law: P (n) = 2−n Event A={outcome is even} Then P (A) = P ({2, 4, 6, . . .}) = P (2) + P (4) + P (6) + · · · 1 1 1 1 = + 4 + 6 +··· = 2 2 2 2 3 M. Chen (IE@CUHK) ENGG2430C lecture 2 17 / 23 Examples Problem 1.11 (Bonferroni’s inequality): Let A and B be two events. Prove that P (A ∩ B ) ≥ P (A) + P (B ) − 1 (Hint: use 1 ≥ P (A ∪ B ) = P (A) + P (B ) − P (A ∩ B )) Can you generalize to the case of n events A1 , A2 , . . . , An ? That is, to show n P (A1 ∩ A2 ∩ · · · ∩ An ) ≥ ∑ P (Ai ) − (n − 1) i =1 M. Chen (IE@CUHK) ENGG2430C lecture 2 18 / 23 Using Probability Model to Analyze Uncertainty Stage1: construct a probability model for the experiment Sample space Ω Probability law Stage2: work on this fully specified probability model Express the events that we are interested in as subsets of Ω Derive probabilities of the events Deduce interesting properties M. Chen (IE@CUHK) ENGG2430C lecture 2 19 / 23 Romeo and Juliet (Example 1.5) Romeo and Juliet have a date at a given time, and each...
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