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# ioe265f11-Lec04 - Lec04 Counting Techniques/Conditional...

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Lec04 - Counting Techniques/Conditional Probability IOE 265 F11 1 1 Counting Techniques and Conditional Probability 2 Topics Counting Techniques Equally Likely Counting N(A)/N Permutations Combinations Conditional Probability

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Lec04 - Counting Techniques/Conditional Probability IOE 265 F11 2 3 Probability Counting Techniques Equally Likely Counting (# outcomes is small) Product Rule for Ordered Pairs Tree Diagrams Permutations Combinations 4 Simple Counting Techniques When various outcomes are “equally likely”, then computing probabilities reduces to counting. P(A) = N(A) / N Let’s define event A as: The Probability of rolling a Seven from the sum of 2 dice; N= 36 outcomes in sample space Outcomes that yield a sum of 7: (1,6) , (6,1), (2,5) , (5,2) , (3,4) , (4,3) N(A) = 6 N = 36 So, P(A) = 6/36 = 1/6 Note: Here N is relatively small However, N may be quite large or outcomes may not be “equally likely”, so we need additional counting rules.
Lec04 - Counting Techniques/Conditional Probability IOE 265 F11 3 5 Product Rule for Ordered Pairs Ordered Pairs: Select Two Objects Object 1 – select n 1 ways Object 2 – select n 2 ways Total Number of pairs: n 1 *n 2 Examples: Roll Two Dice (6 outcomes per die) total pairs: 36 Suppose all vehicles sold in a dealership have 1 engine and 1 transmission from the following options (outcomes): Engine (Object A): 4-cyl, 6-cyl, 8-cyl Transmission (Object B): Automatic, Manual How may total pairs do you have? 6 Tree Diagrams When in doubt, draw a Tree Diagram: List the Ordered Pairs:

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Lec04 - Counting Techniques/Conditional Probability IOE 265 F11 4 7 Tree Diagrams - Continued Note: Tree Diagrams are not required to have common second generation branches. Need common n k size outcomes to use product rule. Example: A production process consists of machining (M) and finishing (F). Suppose you have three machining operations (M1, M2, M3). After completion, products coming out of M1 and M2 feed finishing operations F1, F2, F3. Whereas products coming out of M3 go to F4, F5, F6. How many ordered pairs can we get? M1 M2 M3 F1 F2 F3 F1 F2 F3 F4 F5 F6 8 General Product Rule We may use the general product rule if we keep adding objects to form ordered collections of k-tuples (n 1 * n 2 *..n k ordered collections) Ordered pair ~ 2-tuple Three objects in collection ~ 3-tuple
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ioe265f11-Lec04 - Lec04 Counting Techniques/Conditional...

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