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# 204 Class Note for STAT 22500 with Professor Martin at Purdue

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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 45 views.

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Date Created: 02/06/15
Stat 225 7 Conditional Probability Practice Problems Solutions 021109 1 Your keychain holds six keysiexactly one is your house key Suppose you arrive home after dark and cannot see the keys To open the door you randomly try each key until you get the correct one making sure you do not try the same key twice Find the probability that the third key tried is the house key using a counting techniques This experiment can be visualized as laying out the keys in a random order If the third key is to be the correct one then we only have 5 of the 6 positions free to play with There are 51 ways to arrange the other ve keys out of the total of 61 arrangements so the probability is 5161 16 the multiplication rule Let Ak be the event that the k th key tried is correct k 12 6 Then we want to nd PAf A A3 the other three tries don7t matter Using the multiplication rule we have 13qu m A 6 Ag PAiPA lAiPAslAi 4 56 45 14 16 a symmetry argument The correct key has come up on one of the six tries A little re ection about the approach in part a reveals that there is nothing particularly special about the third positionithat is all six of the possible positions would have the same probability Since there are six equal probabilities that must add to 1 the probability can be nothing but 16 2 Suppose you have two decks of cardsione is a standard deck and the other is a trick deck77 that has all Clubs four of each face value a If you pick a card at random from the trick deck whats the probability of selecting an Ace of Clubs Let A be the event that an Ace of Clubs is drawn and T the event that the card was drawn from the trick deck Out of the 52 cards there are 4 Ace of Clubs so the conditional probability is PAlT 452 Suppose you pick one of the two decks at random and then take a random card from the selected deck What is the probability of selecting an Ace of Clubs Use the total probability formula PA PAlTPT PAlT0PT0 45212 15212 5104 c A friend plays the game in b but does not tell you which deck she chose from7 only that she selected an Ace of Clubs What is the probability she drew from the standard deck We want to nd PTclA which we know would require Bayes formula c PAlTCPTC 15212 Pam PM 7 5104 715 3 The Purdue men7s and women7s basketball teams are both playing on the same night Suppose that the men will win with probability 075 and the women with probability 070 The probability that at least one of them will win is 095 a If the women win7 what is the probability that the men will also win Use the de nition of conditional probability and the addition rule PM m W PM PW i PM o W P M W lt l PM PM 075 070 7 095 0714 070 b Are the events M men win and W women win independent Since 0714 31 075 PM7 the two events are not independent

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