combinatorial_examples

combinatorial_examples - 1.017/1.010 Examples of...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 . If the
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/29/2011 for the course CIVIL 1.00 taught by Professor Georgekocur during the Spring '05 term at MIT.

Page1 / 5

combinatorial_examples - 1.017/1.010 Examples of...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online