Counting - Le arning Obje s ctive fine pling with / without...

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Learning Objectives Define sampling with / without replacement and order / unordered and relate to counting methods Calculate probabilities using the counting principle by selecting the correct method or combination of methods: k-permutations combinations partitions
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Example Polymers are to be created from three kinds subunits. There are 4 subunits of type A, 3 of type B, and 2 of type C. How many different 9-unit polymers are possible?
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Counting Principle Consider a process that consists of r stages. Suppose that: There are n 1 possible results at the first stage For every possible result of the first stage, there are n 2 possible results at the second stage More generally, for any possible results of the first i-1 stages, there are n i possible results at the ith stage Then, the total number of possible results of the r-stage process is: n 1 n 2 …n r Example: roll 2 dice – 6*6=36 possible results
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Exercises Exercise #1 Exercise #2
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Questions to Ask Do we care about the order in which objects are selected? Ordered UnOrdered example: order of the cards you draw in poker doesn’t matter Do we get to reuse objects? With Replacement Without Replacement
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Approaches Ordered UnOrdered With Replacement Without Permutations Combinations n ! ( n - k )! n * n * L = n k n k = n ! k !( n - k n + k - 1 k
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Ordered Sampling With Replacement Care about the order objects are selected When an object is selected it can be re-selected Count possibilities using: select k from n n * n * L = n k
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Example
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This note was uploaded on 02/22/2010 for the course BME 335 taught by Professor Dunn during the Spring '10 term at University of Texas at Austin.

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Counting - Le arning Obje s ctive fine pling with / without...

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