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Tutorial 5: Discrete Probability I

This tutorial covers the formal language of probability, probability space elements, uniform distribution, calculating probabilities, language of probability, and the concept of independence in probability.

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Tutorial 5: Discrete Probability I

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  1. Tutorial 5: Discrete Probability I Steve Gu Feb 15, 2008 Reference:http://www.cs.duke.edu/courses/fall07/cps102/lecture11.ppt

  2. Language of Probability The formal language of probability is a very important tool in describing and analyzing probability distribution

  3. Probability Space • A Probability space has 3 elements: • Sample Space : Ω • All the possible individual outcomes of an experiment • Event Space : ₣ • Set of all possible subsets of elements taken from Ω • Probability Measure : P • A mapping from event space to real numbers such that for any E and F from ₣ • P(E)>=0 • P(Ω)=1 • P(E union F) = P(E) + P(F) • We can write a probability space as (Ω, ₣,P)

  4. Sample Space: Ω Ω 0.13 0.17 0.1 0.11 0.2 0 probability of x 0.1 0.06 0.13 p(x) = 0.2 Sample space

  5.  p(x) PrD[E] = x  E Events Any set E  Ω is called an event S 0.17 0 0.1 0.13 PrD[E] = 0.4

  6. |E| |Ω|  p(x) = PrD[E] = x  E Uniform Distribution If each element has equal probability, the distribution is said to be uniform

  7. A fair coin is tossed 100 times in a row What is the probability that we get exactly half heads?

  8. Using the Language The sample space Ω is the set of all outcomes {H,T}100 Each sequence in Ωis equally likely, and hence has probability 1/|Ω|=1/2100

  9. Visually Ω = all sequences of 100 tosses x = HHTTT……TH p(x) = 1/|Ω|

  10. 100 50 / 2100 Event E = Set of sequences with 50H’s and 50T’s Set of all 2100 sequences{H,T}100 Probability of event E = proportion of E in Ω

  11. Suppose we roll a white die and a black die What is the probability that sum is 7 or 11?

  12. Same Methodology! Ω = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) } Pr[E] = |E|/|Ω| = proportion of E in S = 8/36

  13. 23 people are in a room Suppose that all possible birthdays are equally likely What is the probability that two people will have the same birthday?

  14. Count |E| instead! And The Same Methods Again! Sample space Ω = {1, 2, 3, …, 366}23 x = (17,42,363,1,…, 224,177) 23 numbers Event E = { x  W | two numbers in x are same } What is |E|?

  15. all sequences in S that have no repeated numbers E = |E| = (366)(365)…(344) |E| |E| = 0.494… |Ω| |Ω| |Ω| = 36623 = 0.506…

  16. Pr [ A  B ] Pr [ B ] B A More Language Of Probability The probability of event A given event Bis written Pr[ A | B ] and is defined to be = S proportion of A  B to B

  17. Suppose we roll a white die and black die What is the probability that the white is 1 given that the total is 7? event A = {white die = 1} event B = {total = 7}

  18. Ω = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) } |A  B| Pr [ A  B ] 1/36 Pr [ A | B ] = = = |B| 1/6 Pr[B] event A = {white die = 1} event B = {total = 7}

  19. Independence! A and B are independent events if Pr[ A | B ] = Pr[ A ]  Pr[ A  B ] = Pr[ A ] Pr[ B ]  Pr[ B | A ] = Pr[ B ]

  20. Independence! A1, A2, …, Ak are independent events if knowing if some of them occurred does not change the probability of any of the others occurring Pr[A1 | A2 A3] = Pr[A1] Pr[A2 | A1 A3] = Pr[A2] Pr[A3 | A1 A2] = Pr[A3] E.g., {A1, A2, A3} are independent events if: Pr[A1 | A2 ] = Pr[A1] Pr[A1 | A3 ] = Pr[A1] Pr[A2 | A1 ] = Pr[A2] Pr[A2 | A3] = Pr[A2] Pr[A3 | A1 ] = Pr[A3] Pr[A3 | A2] = Pr[A3]

  21. Silver and Gold One bag has two silver coins, another has two gold coins, and the third has one of each One bag is selected at random. One coin from it is selected at random. It turns out to be gold What is the probability that the other coin is gold?

  22. Let G1 be the event that the first coin is gold Pr[G1] = 1/2 Let G2 be the event that the second coin is gold Pr[G2 | G1 ] = Pr[G1 and G2] / Pr[G1] = (1/3) / (1/2) = 2/3 Note: G1 and G2 are not independent

  23. The Monty Hall Problem • http://www.youtube.com/watch?v=mhlc7peGlGg

  24. The Monty Hall Problem http://en.wikipedia.org/wiki/Monty_Hall_problem

  25. The Monty Hall Problem

  26. The Monty Hall Problem • Still doubt? • Try playing the game online: • http://www.theproblemsite.com/games/monty_hall_game.asp

  27. Thank you! Q&A

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