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combinatorial_examples

combinatorial_examples - 1.017/1.010 Examples of...

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1.017/1.010 Examples of Combinatorial Probability Derivations Example 1 Suppose we toss a die twice. Then the number of possible outcomes is the product of n 1 = 6 outcomes on the first toss and n 1 = 6 outcomes on the second toss, or n 1 n 2 = 36. Now define the event A to correspond to exactly one 3 in 2 tosses. This can occur in two ways. First, we can obtain a 3 on the first toss ( n 1 = 1, because there is only one way to achieve a 3 on the first toss) but not on the second ( n 5 = 5 because there are 5 ways to not obtain a 3 on the second toss), giving n 1 n 2 = 5. Alternatively, we can obtain a 3 on the second toss but not on the first, giving another n 1 n 2 = 5 outcomes, for a total of 10 outcomes in A . Compute P ( A ) from: 278 . 0 36 10 ) ( ) ( ) ( = = = S n A n A P The same result can be obtained by enumerating all 36 outcomes of two die tosses (e.g. on a tree) and then counting the number with exactly one 3 (do this to check the above result). This die toss experiment is an example of sampling with replacement , because outcomes are drawn from the same set of 6 possibilities on each toss (or sub-experiment). Example 2 Suppose you draw 2 letters at random from a set of 4 (say A, B, C, and D). Each successive draw is from a smaller set of possible outcomes (once the A has been drawn it cannot be drawn again and there is one less possible outcome for the second draw). If the order of the letters is relevant, it follows from the product rule that the number of possible letter pairs is: The 12 ordered pairs produced by this experiment are: {AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, DC} More generally, an ordered subset of k objects drawn from a set of n objects is called a permutation of n objects taken k at a time. The number of permutations of n objects taken k at a time is written P k,n : )! ( ! ) 1 )...( 2 )( 1 ( , k n n k n n n n P n k = + = 1
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This number can be used to count simple events that result from sampling without replacement experiments where sampling order is important. The MATLAB function factorial(n) can be used to evaluate P k,n for relatively small n .
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