Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

Session 3

548 views

Published on

  • Be the first to comment

  • Be the first to like this

Session 3

  1. 1. 7/2/2012 Introduction to Probability: Quantitative Methods – I Bangalore [QM 105] Sessions 3-4-5: price range (lakh frequency Review Probability/ practice problems <=10 17 Bayes Theorem 10 to 20 46 Upgrading prior probabilities on Market Share 20 to 30 40 Bayesian vs. Frequentist 30 to 50 47 Puzzles 50 to 75 25 How reliable is RELIABLE? 75 to 100 13 Clear Tone Radios 100 to 135 9 Characteristics of discrete probability distribution > 135 3 Expected value, standard deviation of a random variable Discrete Uniform and Binomial Probability Distributions price range (lakh frequency <=10 17 Area (sq ft) frequency Relative Frequency to 10 to 20 20 to 30 30 to 50 46 40 47 <= 600 600 to 800 15 20 Probability 50 to 75 75 to 100 100 to 135 25 13 9 800 to 1000 35 > 135 3 1000 to 1250 70 1250 to 1500 20 One flat is chosen at random in Bangalore. What is the probability 1500 to 2000 15 that its price is > 2000 25 What is the probability that the chosen flat has area > 2000 sqft? • 10 lakh or less? 0.085 0.315 0.125 • 20 lakh or less? • More than 20 lakh? 0.685 What is the probability that chosen flat has costs no higher than • Between 10 and 100 lakh? 0.855 a crore but has area more than 2000 sqft? 0.125 ×0.94 ? Joint Probability Area (sq ft) <= 600 to 800 to 1000 to 1250 to 1500 to > Margi Area (sq ft) 600 800 1000 1250 1500 2000 2000 nal <= 600 to 800 to 1000 to 1250 to 1500 to > Margin 600 800 1000 1250 1500 2000 2000 al <=10 0.07 0.015 0 0 0 0 0 0.085 <=10 14 3 17 10 to 20 0.005 0.085 0.145 0 0 0 0 0.23 10 to 20 1 17 29 46 20 to 30 0 0 0.03 0.17 0 0 0 0.2 20 to 30 6 34 40 priceprice 30 to 50 0 0 0 0.18 0.055 0 0 0.235 30 to 50 36 11 47 rangerang (lakh) 50 to 75 9 13 3 25 50 to 75 0 0 0 0 0.045 0.065 0.015 0.125 e(lakh 75 to 100 1 12 13 75 to 100 0 0 0 0 0 0.005 0.06 0.065 100 to 100 to 135 0 0 0 0 0 0.005 0.04 0.045 135 1 8 9 > 135 3 3 > 135 0 0 0 0 0 0 0.015 0.015 Marginal 15 20 35 70 20 15 25 Marginal 0.075 0.1 0.175 0.35 0.1 0.075 0.125 1 1
  2. 2. 7/2/2012 Conditional Probability Conditional Probability Given that a flat has more than 2000 sq ft, what is the probability that it costs more than 100 lakh? P[ A and B] P ( A ∩ B) P[ A given B] = P ( A | B) = P[ B ] P( B ) Area > 2000 Price≤100 area ≤ 2000 price> 100 Classical approach: (In) dependence of Events Counting argument Q. A team of 5 members is to be selected randomly • A and B are said to be independent if from 6 gentlemen and 4 ladies. What is the probability – P[A|B] = P[A] that the team will have – or equivalently P[A|B] = P[A|not B] or... – or equivalently P[A and B] = P[A] × P[B] a) No ladies? b) Three gentlemen? Connection between independent and disjoint events c) At most three gentlemen? Assumption: all possible selections are equally likely Problem (Easy) Problem (easy) The HAL Corporation wishes to improve the resistance of its personal computer to disk-drive and keyboard failures. At present, the design of the computer is such thatAt a soup kitchen, a social worker gathers the following data. disk-drive failures occur only one-third as often as keyboard failures. The probabilityOf those visiting the kitchen, 59 percent are men, 32 percent are of simultaneous disk-drive and keyboard failure is 0.05.alcoholics, and 21 percent are male alcoholics. What is the probability that a random male visitor to the kitchen is an alcoholic? (a) If the computer is 80 percent resistant to disk-drive and/or keyboard failure, how low must the disk-drive failure probability be? (b) If the keyboard is improved so that it fails only twice as often as the disk-drive P[alcoholic and male] (and the simultaneous failure probability is still 0.05), will the disk-drive failure P[alcoholic given male]= probability from part (a) yield a resistance to disk-drive and/or keyboard failure P[male] higher or lower than 90 per cent? 0.21 = = 35.59% 0.59 2
  3. 3. 7/2/2012 Solution Use of probability in questionnaire design: How to get an answer w.o. being sure you’ve asked the question • Flip a coin. If you get H answer Q1 ONLY(in yes/no form) If you D → disk-drive failure K → keyboard failure get T answer to Q2 only (in yes/no form) P(K ∩ D) = 0.05 P(K) = 3 P(D) = 3x say • Q1. Is your mother born in May? • Q2. Do you find this class useless? a) P(K ∪ D) =0.2 = x + 3x –0.05 ⇒ x = 0.0625 = P(D) b) If P(K) = 2 P(D) = 0.125, then P(K ∪D) = 0.1375 P[ yes ] − P[Q1] × P[ yes | Q1] and P[ (K ∪D)c ] = 0.8625 < 90% P[ yes | Q 2] = P[Q 2] i.e, the computer is only 86.25% resistant to failure of either type Finding your perfect “match” Which category of students do you No. Criterion % of opposite gender who satisfy belong to? the criterion 1 P(ready in P( not P(not ready P(category students % of ready in 2 category students each each and | not category) catgeory..) ready) 3 category) A 40.00% 0.9 0.1 0.04 12.12% N B 35.00% 0.6 0.4 0.14 42.42% The Chain-rule C 25.00% 0.4 0.6 0.15 45.45% P[ B1 and B2 and B3 ] =P[B1 ] × P[ B2 | B1 ] × P[ B3 | B1and B2 ] p(not (if independent) =P[B1 ] × P[ B2 ] × P[ B3 ] ready) 0.33 Bayes Rule Bayes Rule -- using table P[ A]× P[B | A] events Prior Prob P[B|Ai] P[BAi] P[Ai|B]P[ A | B] = P[PA∩]B] = [B P[Ai] (ii) (iii) =(i)*(ii) (iv) =(iii)/P[B] P[ A ∩ B] + P[ B ∩ Ac ] A1 (i) P[A1] P[B|A1] P[A1B] P[A1| B] A2 P[A2] P[B|A2] P[A2B] P[A2| B] P[ A]× P[ B | A] Ak P[Ak] P[B|Ak] P[AkB] P[Ak| B]= P[ A] × P[B | A] + P[ Ac ]× P[ B | Ac ] Sum 1 P[B] 3
  4. 4. 7/2/2012 Monty Hall Problem or Story of a father ‘Khul Ja SimSim’ A father announces: “I have two children, born• The car is behind one of the 3 doors. three years apart, one of whom is a boy.”• You select door A. What is the probability that the other child is a• Aman Verma (who knows where the car is) boy? opens door B and shows that this is empty. Gives you an option of “switch”.• Should you stick to your initial choice? “Information – the new language of science” by Hans Christian von Baeyer. 4

×