Week 07b - QMA 2006S2 (Probability)

Week 07b - QMA 2006S2 (Probability) - Jessica Chung &...

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Unformatted text preview: Jessica Chung & Derek Hui Week 07 QMA PASS | 2006 S2 | Tues, 3-4pm | QUAD G042 Questions? Email us: [email protected] , [email protected] 1/3 PROBABILITY Probability refers to the likelihood that an event will occur. It is expressed as a ratio, such that results are always between 0 and 1. However, we will begin with the concepts of permutations and combinations – which relate to the way things are counted. PERMUTATIONS A permutation is an ordered arrangement of r objects, without repetition, selected from n distinct objects. The number of permutations is denoted n P r Consider a group of 4 objects: A, B, C and D. To make arrangements, we can begin by putting A in the first position. With no repetitions, this leaves 3 options, or (n – 1) options, for the second position. If we fill this position with B, this leaves 2 options for the remaining positions, or (n – 2) options. 1 2 3 4 r n n – 1 n – 2 n – 3 n – r + 1 This means that, in total, the number of arrangements possible without repetition is given by n(n-1)(n-2) ⋅⋅⋅ (n-r+1), etc. To calculate permutations more easily, we can use factorials. ! ( 1)( 2) ( 1) ( )! n r n P n n n n r n r =-- ⋅⋅⋅- + =- EXAMPLE A Find the number of permutations of 4 elements taken 4 at a time. 4! 4 3 2 1 24 1! 1 n r P × × × = = = EXAMPLE B How many ways can you seat seven people in a row of three seats? 7! 7 6 5 4 3 2 1 210 (7 3)! 4 3 2 1 n r P × × × × × × = = =- × × × COMBINATIONS A combination is the number of possible selections of ‘r’ elements from a set of ‘n’ elements without regard to order . Consider a group of 4 objects: A, B, C and D. To choose groups of four, we can consider how many permutations are possible. From Example B above, we know that there are 24 possible permutations. However , note that ABCD and ABDC and ACDB ... etc ... are essentially the same group of objects....
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This note was uploaded on 04/02/2012 for the course ECON 1101 taught by Professor Julia during the Three '08 term at University of New South Wales.

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Week 07b - QMA 2006S2 (Probability) - Jessica Chung &...

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