DARTMOUTH COSC 030 - 08prob (4 pages)

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Experiments With Random Outcomes Examples Toss a fair coin which side does it land on CS 30 Discrete Mathematics Outcomes HEADS TAILS Roll a fair die what number shows on top Outcomes 1 2 3 4 5 6 ShufOle a deck of cards pick top which card Amit Chakrabarti Probability Probability Spaces Sample space set of outcomes of an experiment Any nonempty set can be a sample space Ours in this course will usually be Oinite sets Sometimes we ll use N or N 0 or Z as a sample space Probability function on such a sample space S is a function Pr S R such that nonnegativity x S Pr x 0 normalization x S Pr x 1 Probability space a pair S Pr Outcomes 2 2 7 Q A Toss a biased coin heads 3 as likely as tails Outcomes HEADS TAILS but something s different Sort n numbers using QuickSort how fast does it run Train a deep belief network on images of puppies Some Simple Probability Spaces Some uniform probability spaces Toss a fair coin which side does it land on Take S H T Pr H 1 2 Pr T 1 2 Roll a fair die what number shows on top Take S 1 2 3 4 5 6 Pr 1 Pr 2 Pr 6 1 6 ShufOle a deck of cards pick top which card Take S 2 2 7 Q A Pr x 1 52 for each x S A nonuniform probability space Toss a biased coin heads 3 as likely as tails Take S H T Pr H 3 4 Pr T 1 4 Modeling it Right How is the Pr function determined It s a modeling choice Roll two fair dice number thrown sum of values Possible outcomes S 2 3 4 12 is Pr uniform Correct question Should we de ine Pr to be uniform over S If we did then Pr 7 1 11 Let s try it out Uh oh looks like Pr 7 1 6 how to justify Model it right an outcome is a pair die1 die2 So S1 1 1 1 2 1 3 6 6 If dice are fair it makes sense to let Pr be uniform over S1 Sum 7 when outcome is one of 1 6 2 5 3 4 4 3 5 1 6 1 Pr Sum 7 1 36 1 36 1 36 1 36 1 36 1 36 1 6 Uniform Probability Space Suppose S Pr is a uniform probability space For all x S Pr x 1 S For all E S Pr E x E Pr x E S Easy calculations Roll two fair dice Pr throwing a 7 Take S 1 1 1 2 1 3 6 6 uniform probabilities Event of interest E7 1 6 2 5 3 4 4 3 5 2 6 1 Pr E7 E7 S 6 36 1 6 Once you re a consummate expert Pr throwing a 7 6 favorable