lecture09

Lecture09 - COMPSCI 102 Discrete Mathematics for Computer Science Lecture 9 Probability Theory Counting in Terms of Proportions The Descendants of

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Unformatted text preview: COMPSCI 102 Discrete Mathematics for Computer Science Lecture 9 Probability Theory: Counting in Terms of Proportions The Descendants of Adam Adam was X inches tall He had two sons: One was X+1 inches tall One was X-1 inches tall Each of his sons had two sons … X X-1 X+1 X-2 X+2 X X-3 X+3 X-1 X+1 X-4 X+4 X-2 X+2 X 1 1 1 1 1 2 1 3 3 1 1 4 6 4 1 In the n th generation there will be 2 n males, each with one of n+1 different heights: h , h 1 ,…,h n h i = (X-n+2i) occurs with proportion: n i / 2 n Unbiased Binomial Distribution On n+1 Elements Let S be any set {h , h 1 , …, h n } where each element h i has an associated probability Any such distribution is called an Unbiased Binomial Distribution or an Unbiased Bernoulli Distribution n i 2 n 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 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? 3 choices of bag 2 ways to order bag contents 6 equally likely paths Given that we see a gold, 2/3 of remaining paths have gold in them! So, sometimes, probabilities can be counterintuitive ?...
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Lecture09 - COMPSCI 102 Discrete Mathematics for Computer Science Lecture 9 Probability Theory Counting in Terms of Proportions The Descendants of

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