Lesson7S1_CS.pptx - TAIBAH UNIVERSITY Faculty of...

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TAIBAH UNIVERSITY Faculty of Science Department of Math . ةبيط ةعماج مولعلا ةيلك تايضايرلا مسق Probability and Statistics for Engineers STAT 305 Teacher : Dr. Moustapha Abdellahi First Semester 1438/1439
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Counting Techniques Lesson 7
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Contents Multiplication Rule Permutations Combinations
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Counting Sample Points: There are many counting techniques which can be used to count the number points in the sample space (or in some events) without listing each element. In many cases, we can compute the probability of an event by using the counting techniques.
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Multiplication Rule: If an operation can be performed in n 1 ways , and if for each of these ways a second operation can be performed in n 2 ways , then the two operations can be performed together in n 1 n 2 ways. Theorem
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Multiplication Rule (Example 1): How many sample points are: there: in the sample space when a pair of dice is thrown once?
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Multiplication Rule (Example 1): The first die can land in any one of n 1 =6 ways. For each of these 6 ways the second die can also land in n 2 =6 ways. Therefore, the pair of dice can land in: n 1 n 2 = (6)(6) = 36 possible ways .
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Multiplication Rule (Example 1):
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Multiplication Rule: If an operation can be performed in n 1 ways, and if for each of these a second operation can be performed in n 2 ways, and for each of the first two a third operation can be performed in n 3 ways, and so forth, then the sequence of k operations can be performed in n 1 n 2 …….. n k ways. Theorem
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Multiplication Rule (Example 2): Sam is going to assemble a computer by himself. He has the choice of ordering chips from two brands, a hard drive from four, memory from three, and an accessory bundle from five local stores. How many different, ways can Sam order the parts?
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Multiplication Rule (Example 2): Since n 1 = 2 , n 2 = 4 , n 3 = 3 and n 4 = 5 There are : n 1 × n 2 × n 3 × n 4 = 2 × 4 × 3 × 5 = 120 different ways to order the parts. Solution:
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Multiplication Rule (Example 3): How many even four-digit numbers can be formed from the digits 0, 1, 2, 5, 6, and 9 if each digit can be used only once?
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Multiplication Rule (Example 3): Since the number must be even, we have only n 1 = 3 (0,2,6) choices for the units position Hence we consider the units position by two parts, 0 or not 0 .
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