STOR 435 February 16 2009 Name Old Midterm Exam 1 Key All problem parts have equal weight In budgeting your time expect that some problems will take longer than others The exam is closed book and closed note You will only need a calculator and a pen or pencil If you need additional paper for solutions or work please ask the instructor To receive any or partial credit for a question you must show your work Please give your answers in the space provided All numerical answers should be expressed as decimal numbers rather than fractions When you have nished please sign the following Honor Code pledge I pledge that I have neither given nor received unauthorized assistance during this examination Signed 1 Let us assume that the events A1 Ak are independent Prove or l 1Al 1 cid 81 k l 1 1 P Al disprove P k Solution P k l 1Al 1 P k 1 P k cid 123 l 1Al cid 123 l l 1A 1 k cid 89 1 k cid 89 l 1 l 1 cid 123 l P A 1 P Al The second equality is due to the De Morgan s laws and the third due to independence 2 A fair die is tossed n times What is the probability of the event at east one face never appears Solution Denote Ak the evenet that the face k does not appear We need to calculate P A1 A6 Using the inclusion and exclusion principle and symmetry we get 6 cid 91 j 1 P Aj 6 cid 88 P Ai 6 cid 88 cid 18 6 cid 19 4n cid 19 5n cid 18 6 6n i 1 2 1 1 i j 6 6n 6 cid 88 cid 18 6 cid 19 2n 1 i j k 6 cid 18 6 cid 19 1 6n 6n 5 P Ai Aj cid 18 6 cid 19 3n 6n 3 4 P Ai Aj Ak 3 Jan s big brown dog Shtutzy knows how to open the fridge One day Jan leaves a dozen 12 eggs in his fridge Two of the eggs are rotten the rest is good When Jan comes home the fridge is ransacked Among other things Shtutzy ate 5 eggs out of the dozen Assume that she picked the eggs at random without paying any attention to whether the eggs are good or not a What is the probability that Shtutzy ate exactly one rotten egg b Shtutzy is known to have a stomach of steel If she eats one rotten egg she will be sick with probability 0 2 If she eats two rotten eggs she will be sick with probability 0 5 If she does not eat any rotten egg she still might be sick from eating too much with probability 0 01 What is the chance Shtutzy will be sick c I have a good news Shtutzy was not sick Jan wanted to have eggs next morning for breakfast Since he was sleepy he picked three out of the remaining eggs at random What is the chance none of the three eggs is rotten Hint This is a conditional probability Solution We will denote the following events by Ri Shtutzy ate i rotten eggs i 0 1 2 S Shtutzy got sick J Jan had no rotten egg a P R1 530 1 10 2 4 12 5 b P S P S R0 P R0 P S R1 P R1 P S R2 P R2 cid 1 cid 0 10 cid 0 2 cid 1 cid 0 12 cid 1 0 2 0 5 cid 1 cid 0 10 cid 0 2 cid 1 cid 0 12 cid 1 0 5 4 1 cid 1 cid 0 10 cid 0 2 cid 1 cid 0 12 cid 1 0 185 3 2 0 01 5 5 5 c P J S cid 123 cid 123 P J S P S cid 123 P J S cid 123 R0 P R0 cid 123 R2 P R2 cid 123 R0 P S 1 P S cid 123 R2 P S 1 P S 3 6 0 10 2 5 1 10 2 4 0 8 5 3 5 12 7 12 1 0 185 P J S 0 99 5 3 3 7 P J S cid 123 R1 P S 1 P S cid 123 R1 P R1 0 5 7 3 3 7 2 10 2 3 5 12 0 501 4 Let X be a random variable with probability mass functions x p x 1 1 4 0 1 2 2 1 4 Find the following a EX b Var X c E 2X d P X 3 1 4 0 1 Solution a EX 1 1 b Var X EX 2 EX 2 1 2 1 c E 2X 2 1 1 d P X 3 1 1 4 20 1 4 1 2 22 1 2 3 2 2 1 4 13 4 1 4 4 8 4 02 1 2 22 1 4 1 4 2 19 16
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