BCR, Permutation, Combination, and M.C

BCR, Permutation, Combination, and M.C - BCR Basic Counting...

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Unformatted text preview: BCR Basic Counting Rule Tree Diagrams etc. Suppose we have the following situation: An assiduous student named Sam finds himself hungry at 2 a.m. on a Tuesday. This unremitting undergraduate has become conscious of a considerable craving for Connie's pizza. Alas, he is nowhere near Chicago to fulfill such a phenomenal food fantasy. He has to settle for 1 of 5 pizza places that still permits pie purchases. Each restaurant has 3 choices for crust type: thin crust, regular, and deep dish. Additionally, a customer is allowed to have at most 1 meat out of the 4 total choices and at most 1 vegetable out of the 5 for his toppings. How many possible pizzas could the famished freshman feast on? Ceteris paribus, how many possible pizzas are there if the place permits at most 2 meat choices? Ceteris paribus, how many pizzas are possible if the restaurant allows at most 2 meat choices and any combination of vegetables? Another Scenario: A young man wants to plan a nice date for his girlfriend. He has the option of going to Chicago, Indianapolis, or staying Lafayette. If he chooses Chicago, he has 10 choices for a play and 100 choices for a restaurant. If he opts of Indianapolis, he has 5 choices for a play and 50 choices for a restaurant. If he remains in Lafayette, he only has 2 options for a play 20 choices for a restaurant. How many options does this gentleman have for a romantic evening out? What about if we added the option of Fort Wayne and this particular city boasted 8 plays and 75 restaurants? BCR suppose that r actions (choices) are to be performed in a definite order. Further suppose that there are m1 possibilities for the 1st action, m2 possibilities for the 2nd action, etc. Then there are m1 * m2 *...* mr possibilities altogether for the r actions. Example: Illinois license plates consist of 4 digits followed by 2 letters. Whereas, in Ohio license plates start with 3 letters and end with 4 digits. a) For each state, how many possible license plates are there? b) How many possible license plates are possible for each state if no digit or letter is allowed to repeat? The above is a simple example of sampling with replacement versus sampling without replacement. This CONCEPT IS CRUCIAL IN PROBABILITY. These phrases alter probabilities. Sometimes the difference is quite large, other times they are practically negligible. However, for our purposes, a difference is still a difference no matter how small. Important Symbols: N is population size (sampling) Sampling with replacement: a) How many possible ordered samples of size 3 with replacement from a population of size 4? b) In general, how many possible ordered samples of size n with replacement from a population of size N. PERMUTATIONS AND COMBINATIONS Factorial the product of the 1st k positive integers is called k factorial, denoted k!. k!= k*(k1)*(k2)*...*1 6! = 6*5*4*3*2*1 = 720 and n is sample size Permutation a permutation of r objects from a collection of m objects is any ORDERED arrangement of r distinct objects from the m objects. Simple Example: Suppose there are 20 colleagues that are trying to form a committee of 4 people. The 4 people are respectively the project manager, chief organizer, treasurer and the secretary. How many possible ways are there to choose this particular committee? What about if we had the 4 above position and 3 positions with no associated titles? What about if we just wanted a committee of 4 members from the original 20 with no distinction between the 4 members? A permutation is denoted by ( m ) r and this can be computed numerically as follows: ( m) r = m! = m * (m - 1) * (m - 2) *...* (m - r + 1) . (m - r )! For the 1st part of the committee example we have (20) 4 = 20 *19 *18 *17 . So, another way to look at ( m ) r is that it is the first r components of m factorial. Special Permutation Rule (m) m = m ! Permutations can be applied to situations of sampling without replacement. Consider the set A={1,2,3,...,20}. Suppose we wanted to draw a sample of size 5 from this set. If the ORDER of the sample matters, then this is an example of a permutation. Order matters implies that {1,2,3,4,5} is not the same as {5,1,2,3,4}. However, if order does not matter then it is no longer a permutation. Then, it would be a combination. Combination a combination of r objects from a collection of m objects is any UNORDERED arrangement of r objects from the m objects. The 3rd and final part of the committee example is an example of a combination. Namely, if we wanted a committee of 4 members from 20 with no distinction between the 4 members, then we have a combination. How do we determine the number of possibilities numerically? m Well, a combination of r objects from a collection of m objects is written as . This is computed as r m m! follows: = . r r !* (m - r )! You can see that the numerical difference between a permutation and a combination is the r! in the denominator of the combination. Therefore, a combination is always smaller than or equal to a permutation. This r! is because there are r! ways that a set of r elements can be ordered. For a basic example think about the set {1,2,3}. You can have {1,2,3}, {1,3,2}, {2,1,3}, {2,3,1}, {3,1,2}, and {3,2,1}. As you can see this is 6 sets if order matters. If order does not matter then all 6 sets are equivalent. This is the reason for the addition of r! in the denominator. m A note on terminology: are referred to as binomial coefficients for their role in the probability r mass function of the Binomial distribution. Poker Example: Using a standard 52 card deck, how many possible ways are there to get a 5 card poker hand? Using a standard 52 card deck, how many possible ways are there to get a 7 card poker hand? Using a standard 52 card deck, how many possible ways are there to get your hold cards in Texas Hold'em? Using a standard 52 card deck, how many possible ways are there to get your hold cards in Omaha? How many possible ways are there to a 3 of a kind in a standard game of 5 card poker? What is the probability of a 3 of a kind? If you were playing Spades or Hearts, how many possible ways would there be for you to get your cards? If you were playing Spades or Hearts, how many possible ways would there be for your team to get your cards? If you were playing Spades or Hearts, how many possible ways would there be to deal out all the cards? Ordered Partition An ordered partition of m objects into k distinct groups of sizes m1 , m 2 , ... , m k is any division of the m objects into a combination of m1 objects constituting the first group, m 2 objects constituting the second group, etc. The number of such partitions possible is denoted by m m m! . . This can be computed by = m1 , m 2 , ... , m k m1 , m 2 , ... , m k m1 !* m2 !*...* mk ! Estoy tan cansado; as renuncio ahorita. ...
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