sol1A[1]

sol1A[1] - Larget Statistics/Mathematics 309 Exam 1A...

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Unformatted text preview: Larget Statistics/Mathematics 309 Exam 1A Solution October 3, 2008 A fair 8-sided die (with numbers 1, 2, . . . , 8) is rolled consecutively seven times, and the values rolled are recorded. The die rolls are mutually independent. Solution: The sample space is the set of outcomes S = { ( x 1 , . . . , x 7 ) : x i { 1 , 2 , . . . , 8 }} which has size | S | = 8 7 and each outcome is equally likely. 1. Find the probability that there are exactly three 1s and three 2s rolled. Solution: A good strategy to solve this problem is to first count the possible choices for the numbers and then to multiply this count by the possible number of orders for each. There are seven die rolls. The event is that there are exactly three 1s and three 2s, so there is only one other number to determine. This number cannot be a 1 and a 2, so there are six choices left. If this value is x , then the numbers on the dice are 1 , 1 , 1 , 2 , 2 , 2 , x . There are many ways to count the number of ways to arrange these value in a sequence. The multinomial coefficient ( 7 3 3 1 ) counts directly the number of ways to partition the seven positions into three groups of size three, three, and one. Alternatively, we could firstways to partition the seven positions into three groups of size three, three, and one....
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sol1A[1] - Larget Statistics/Mathematics 309 Exam 1A...

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