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# Sol IV - ENGM2032 Applied Probability and Statistics Winter...

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ENGM2032, Applied Probability and Statistics Winter 2010 Assignment #4: Solutions 1) a) We use the counting principles of Chapter 2 to determine the probability mass function (pmf) of X . That is, we think of the deck of cards as being composed of two groups, the 13 spades and the 39 ‘other’ cards. Thus the event X = i means that we have selected i cards from the 13 spades and 4 i cards from the 39 other cards. The desired probabilities are P X = 0 = 13 0 39 4 52 4 = 39 × 38 × 37 × 36 4 × 3 × 2 × 1 52 × 51 × 50 × 49 4 × 3 × 2 × 1 = 82 , 251 270 , 725 = 0 . 30382 P X = 1 = 13 1 39 3 52 4 = 0 . 43885 P X = 2 = 13 2 39 2 52 4 = 0 . 21349 P X = 3 = 13 3 39 1 52 4 = 0 . 04120 P X = 4 = 13 4 39 0 52 4 = 0 . 00264 As a check on our math, we note that 0 . 30382+0 . 43885+0 . 21349+0 . 04120+0 . 00264 = 1, as it should. b) To get the expected value of X , we compute E X = 4 x =0 x P X = x = 0(0 . 30382)+1(0 . 43885)+2(0 . 21349)+3(0 . 04120)+4(0 . 00264) = 1 which makes sense because 1 / 4 of the cards are spades, so one would expect one spade every 4 cards, on average. 2) a) P X > 700 = 700 0 . 002 e - 0 . 002 x dx = e - 0 . 002 x 700 = e - 0 . 002(700) = 0 . 2466 b) We have 7 3

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