180amid1revsolns - 180 Midterm 1 Review Solutions 1 If 8...

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180 Midterm 1 Review Solutions October 10, 2006 1) If 8 castles (i.e. rooks) are randomly placed on a chessboard, what are the chances that no rook can capture any of the others? In other words, what is the probability that no row or column contains two or more rooks? Answer We first note that the standard chessboard is an 8 × 8 grid, hence it has 64 squares, and there are ( 64 8 ) different configurations of placing 8 rooks on the board. How many possibilities are there where no rook can capture another? There must be exactly one rook in each column. We note that we can put the rook in the first column in any of the 8 rows. The next rook has to be in a different column, and thus has 7 choices, and so on. Thus there are 8! configurations with all rooks safe. This gives us the probability: 8! ( 64 8 ) 2) Two dice one red, the other blue are rolled. What is the probability that the red die lands on a higher number than the blue die. Answer One can simply count the possibilities and get 15 36 As an alternate method one can note that the probability that both dice rolls are the same is 1 / 6 . Also, note that the probability that the red die is higher is the same as the probability that the blue die is higher. Thus the probability is 1 - 1 / 6 2 = 15 36 3) A woman has n keys that all look alike, of which one will open her door. (a) If she tries the keys at random, discarding those that do not work, what is the probability that she will open the door on her k th try?
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