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math 105 Prelim2 +key 2007

math 105 Prelim2 +key 2007 - MATH 105 Prelim 2 and...

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MATH 105 Prelim 2 and Solutions- Fall 2007 Problem 1 - (15 total points) Consider an experiment where you roll two fair dice and add the two values together. (a) (5 points) Write down the sample space of the experiment above. Solution: All possible sums of 2 dice give S = { 2 , 3 , ..., 12 } . (b) (5 points) Show that the the sample space is not equally likely by computing the probabilities of the outcomes 12, 11 and 10. Solution: Looking at the outcome of the possible rolls of two dice, we see that there is 1 way to get a 12, 2 ways to get an 11 (5,6 or 6,5), and 3 ways to get a 10 (4,6; 5,5 or 6,4). Since there are 6x6 equally likely outcomes (pairs of numbers) that are possible, we have that P (10) = 3 36 = 1 13 P (11) = 2 36 = 1 18 P (12) = 1 36 (c) (5 points) What is the probability of getting a sum which is less than or equal to 9? Solution: P ( 9) = 1 - P ( 10) = 1 - P (10) - P (11) - P (12) = 1 - 3 36 - 2 36 - 1 36 = 5 6 Problem 2 - (17 total points) A class has 10 boys and 12 girls. People in the class have either long or short hair. 25% of the girls have short hair and 20% of the boys have long hair. Calculate the following probabilities. Solution: Note that the following problems are made easier by first computing the probability (or count) of the 4 “intersections”: “short haired boy”, “short haired girl”, “long haired boy”, and “long haired girl” (which are 8, 3, 2, and 9 respectively). (a) (5 points) The probability that a person from the class has long hair?
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