Tutorial 5: Discrete Probability IReference:http://www.cs.duke.edu/courses/fall07/cps102/lecture11.pptSteve GuFeb 15, 2008Language of ProbabilityThe formal language of probability is a very important tool in describing and analyzing probability distributionProbability 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)ΩSample spaceSample Space: Ωp(x) = 0.2probability of x0.20.130.060.110.170.10.1300.1EventsAny set E  Ω is called an event p(x)x  EPrD[E] = S0.170.10.130PrD[E] = 0.4Uniform DistributionIf each element has equal probability, the distribution is said to be uniform p(x) = x  EPrD[E] = |E||Ω|A fair coin is tossed 100 times in a rowWhat is the probability that we get exactly half heads?The sample space Ω is the set of all outcomes {H,T}100Each sequence in Ω is equally likely, and hence has probability 1/|Ω|=1/2100Using the LanguageΩ = all sequencesof 100 tosses x = HHTTT……THp(x) = 1/|Ω|VisuallySet of all 2100sequences{H,T}100Probability of event E = proportion of E in ΩEvent E = Set of sequences with 50H’s and 50 T’s10050/ 2100Suppose we roll a white die and a black die What is the probability that sum is 7 or 11?(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 = 8/36Same Methodology!Ω = {23 people are in a roomSuppose that all possible birthdays are equally likelyWhat is the probability that two people will have the same birthday?x = (17,42,363,1,…, 224,177)23 numbersAnd The Same Methods Again!Sample space Ω = {1, 2, 3, …, 366}23Event E = { x  W | two numbers in x are same }Count |E| instead!What is |E|?all sequences in S that have no repeated numbersE =|Ω| = 36623|E| = (366)(365)…(344)= 0.494…|Ω||E||E||Ω|= 0.506…and is defined to be = SABproportion of A  BMore Language Of ProbabilityThe probability of event A given event B is written Pr[ A | B ]to BPr [ A  B ]Pr [ B ]event A = {white die = 1}event B = {total = 7}Suppose we roll a white die and black dieWhat is the probability that the white is 1 given that the total is 7?(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) }Ω = {|B|Pr[B]1/6|A  B|=Pr [ A | B ]Pr [ A  B ]1/36= =event A = {white die = 1} event B = {total = 7}Independence!A and B are independent events ifPr[ A | B ] = Pr[ A ]Pr[ A  B ] = Pr[ A ] Pr[ B ] Pr[ B | A ] = Pr[ B ]E.g., {A1, A2, A3}are independent events if:Pr[A1| A2 A3] = Pr[A1]Pr[A2| A1 A3] = Pr[A2]Pr[A3| A1 A2] = Pr[A3]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]Independence!A1, A2, …, Akare independent events if knowing if some of them occurred does not change the probability of any of the others occurringSilver and GoldOne bag has two silver coins, another has two gold coins, and the third has one of eachOne bag is selected at random. One coin from it is selected at random. It turns out to be goldWhat is the probability that the other coin is gold?Let G1be the event that the first coin is goldPr[G1] = 1/2Let G2be the event that the second coin is goldPr[G2| G1] = Pr[G1and G2] / Pr[G1]= (1/3) / (1/2)= 2/3Note: G1and G2are not independentThe Monty Hall Problem• http://www.youtube.com/watch?v=mhlc7peGlGgThe Monty Hall Problemhttp://en.wikipedia.org/wiki/Monty_Hall_problemThe Monty Hall ProblemThe Monty Hall Problem• Still doubt?• Try playing the game online:– http://www.theproblemsite.com/games/monty_hall_game.aspThank