lecture09 - Probability Theory I(Flipping Coins for...

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1 Lecture 9 (September 27, 2011) Probability Theory I ( Flipping Coins for Computer Scientists) Some Puzzles Teams A and B are equally good In any one game, each is equally likely to win What is most likely length of a “best of 7” series? Flip coins until either 4 heads or 4 tails Is this more likely to take 6 or 7 flips? 6 and 7 Are Equally Likely To reach either one, after 5 games, it must be 3 to 2 ½ chance it ends 4 to 2; ½ chance it doesn’t Teams A is now better than team B The odds of A winning are 6:5 What is the chance that A will beat B in the “best of 7” world series? i.e., in any game, A wins with probability 6/11 Silver and Gold A 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?
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2 Let us start simple… A fair coin is tossed 100 times in a row What is the probability that we get exactly 50 heads? The set of all outcomes is {H,T} 100 There are 2 100 outcomes Out of these, the number of sequences with 50 heads is 100 50 / 2 100 100 50 If we draw a random sequence, the probability of seeing such a sequence: = 0.07958923739… The sample space S , the set of all outcomes, is {H,T} 100 The Language of Probability “A fair coin is tossed 100 times in a row” Each sequence in S is equally likely, and hence has probability 1/|S|=1/2 100 “What is the probability that we get exactly 50 heads?” Let E = {x in S| x has 50 heads} be the event that we see half heads. The Language of Probability Pr(E) = |E|/|S| = |E|/2 100 Pr(E) = x in E Pr(x) = |E|/2 100 Set S of all 2 100 sequences {H,T} 100 Probability of event E = proportion of E in S Event E = Set of sequences with 50 H ’s and 50 T ’s 100 50 / 2 100
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3 A fair coin is tossed 100 times in a row What is the probability that we get 50 heads in a row ? The sample space S , the set of all outcomes, is {H,T} 100 formalizing this problem… again, each sequence in S equally likely, and hence with probability 1/|S|=1/2 100 Now E = {x in S| x has 50 heads in a row} is the event of interest. What is |E|? 2 50 50 HH  H anything HH H anything T 2 49 HH H T 2 49 HH H T 2 49 HH H T 2 49 49 100 51 52 2 52 0 22 E Total = 50 x 2 49 +2 50 = 52 x 2 49 If we roll a fair die, what is the probability that the result is an even number? ½, obviously True, but let’s take the trouble to say this formally. Each outcome x in S is equally likely, i.e., x in S, the probability that x occurs is 1/6 . sample space S = {1,2,3,4,5,6} P( ) 1 1 6 1 2 6 1 3 6 1 4 6 1 5 6 1 6 6 xx   2,4,6 E 1 1 1 3 1 P( ) 6 6 6 6 2 E Suppose that a dice is weighted so that the numbers do not occur with equal frequency.   E 2 1 3 4 2 P( ) 6 12 12 6 3 E P( ) 1 1/ 6 2 2 / 6 3 1/12 4 5 6 3/12 table of frequencies (proportions) (probabilities)
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4 Language of Probability The formal language of probability is a crucial tool in describing and analyzing problems involving probabilities… and in avoiding errors, ambiguities, and fallacious reasoning.
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lecture09 - Probability Theory I(Flipping Coins for...

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