Chen iecuhk engg2430c lecture 2 14 23 properties of

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ([email protected]) ENGG2430C lecture 2 14 / 23 Properties of Probability Law (from the textbook) M. Chen ([email protected]) 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 ([email protected]) 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 ([email protected]) 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 ([email protected]) 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 ([email protected]) ENGG2430C lecture 2 19 / 23 Romeo and Juliet (Example 1.5) Romeo and Juliet have a date at a given time, and each...
View Full Document

This document was uploaded on 03/31/2014.

Ask a homework question - tutors are online