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CS723 - Probability and Stochastic Processes

CS723 - Probability and Stochastic Processes. Lecture No. 07. In Previous Lectures.

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CS723 - Probability and Stochastic Processes

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  1. CS723 - ProbabilityandStochastic Processes

  2. Lecture No. 07

  3. In Previous Lectures • Application of probability theory to simple combinatronics problems• Discussion limited to problems with finite sample space • Unconditional probability versus conditional probability • Total probability, a-priori probability, and • a-posterior probability• Bayes’s theorem and its applications

  4. Bayes Rule: Application In Lahore, what is the probability that a certain person owns a house in DHA B1 = The person is a civilian =99/100B2 = The person is an army officer=1/100A = The person owns a house in DHAPr(A) = Pr(A|B1) Pr(B1) + Pr(A|B2) Pr(B2) = (0.001) (0.99) + (0.85) (0.01) = 1%Pr(B2|A) = (0.0085) / (0.00949) = 89.6%

  5. Bayes Rule: Application

  6. Random Orientation Angle lies between [0,2π) hence S = {w ε R, 0 ≤ w < 2π}All outcomes are equally likely but there are uncountably infinite outcomesIndividual outcomes are not consideredOnly meaningful events are intervals with non-zero lengthProbability of an interval is proportional to the length of the interval

  7. Partitioned Footpath A footpath with glass strips every 75cmThe strips are 0.5cm thick your show is 30cm long.

  8. Partitioned Footpath S = {d ε R, 0 ≤ d < 75}The foot touches if 44.5 < d < 75 Pr(touching the strip) = 30.5/75 = 40.67%

  9. Ten Pin Bowling Diameter of the ball is 9.6”Diameter of the pin is 4.5”

  10. Partitioned Floor A footpath with glass strips every 75cmThe strips are 0.5cm thick

  11. Partitioned Floor S = {(x,y) ε R2 , 0 ≤ x < p and 0 ≤ y < p} Pr(disk not touching a line) = (d-p)2 / d2 Pr(disk touching a line) = 1 – (d-p)2 / d2

  12. Trains on a Junction Train on North-south arrives every 12 minutes and stops for 2 minutesTrain on East-west line arrives every 13 minutes and stops for 3 minutesProbability that NS train will arrive within 2 minutes of EW arrivalProbability that NS train will arrive after EW train and will depart before itProbabilities of other complex events.

  13. Trains on a Junction

  14. EW Arrives before NS

  15. Buffon’s Needle

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