Probability For Computer Scientists Cas-Books Pdf

Probability for Computer Scientists CAS
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Permutations and Combinations,Suppose that we have n objects O1 O2 On. A permutation of order k is an ordered selection of k of these for 1 k n. A combination of order k is an unordered selection of k of these. Common notation P n k or Pkn n n 1 n k 1 n k, Example Given the 5 letters a b c d e how many ways can we list 3 of the 5. when order is important,Answer P35 53 5 4 3 60, Note that each choice of 3 letters such as a c e results in 6 different results. ace aec cae cea eac eca, Example Given the 5 letters above how many ways can we choose 3 of the 5. when order is NOT important,5 53 5 4 3, Answer 10 In this case we have the 60 that result when we care about order.
divided by 6 the number of orderings of 3 fixed letters. Applied Statistics Probability 1 2,Definition, n k n n 1 n k 1 for any positive integer n and for integers k. such that 1 k n This symbol is pronouced n to the k falling. Examples 6 3 6 5 4 120,36 is not defined for our purposes. Again P n and,Applied Statistics Probability 1 3,Permutations of Multiple Types. The number of permutations of n n1 n2 nr objects of which n1 are of. one type n2 are of a second type and nr are of an rth type is. Example Suppose we have 2 red buttons 3 white buttons and 4 blue buttons. How many different orderings permutations are there. Answer 1260,Applied Statistics Probability 1 4, There are 12 marbles in an urn 8 are white and 4 are red The white marbles. are numbered w1 w2 w8 and the red ones are numbered r1 r2 r3 r4. For a d Without looking into the urn you draw out 5 marbles. a How many unique choices can you get if order matters 12 5 95 040. b How many unique choices can you get if order does not matter 792. c How many ways can you choose 3 white marbles and 2 red marbles if. order matters You will fill 5 slots by drawing First determine which. two slots positions will be occupied by 2 red marbles 10 Next. multiply by orderings of 3 white and 2 red 10i8 3 i4 2 40 320. d How many ways can you choose 3 white marbles and 2 red marbles if. order does not matter 336, e How many marbles must you draw to be sure of getting two red ones 10.
Applied Statistics Probability 1 5,Complex Combinations. How many ways are there to create a full house 3 of a. kind plus a pair using a standard deck of 52 playing. 13i4i12i6 3 744,choose denomination x choose 3 of 4 of given. denomination x choose one of the remaining,denominations x choose 2 of 4 of this second. denomination, This follows from the multiplication principle Theorem 2 3 1 in text. Applied Statistics Probability 1 6,Suppose What is n.
Suppose What is r,Applied Statistics Probability 1 7. Consider a machining operation in which a piece of sheet metal. needs two identical diameter, holes drilled and two identical size notches cut We denote a. drilling operation as d and a, notching operation as n In determining a schedule for a machine. shop we might be interested, in the number of different possible sequences of the four. operations The number of possible, sequences for two drilling operations and two notching operations.
The six sequences are easily summarized ddnn dndn dnnd nddn. Applied Statistics and Probability for Engineers Douglas C Montgomery. George C Runger John Wiley Sons Inc 2006,Applied Statistics Probability 1 8. A printed circuit board has eight different locations in which a. component can be placed, If five identical components are to be placed on the board how. many different designs are possible, Each design is a subset of the eight locations that are to contain. the components The number of possible designs is therefore. 8 8 83 8 7 6,5 3 3 3 2 1, Applied Statistics and Probability for Engineers Douglas C Montgomery. George C Runger John Wiley Sons Inc 2006,Applied Statistics Probability 1 9.
Sample Space,Definition The totality of the possible. outcomes of a random experiment is called,the Sample Space. Outcome from one roll of one die 1 2 3 4 5 6, The number of attempts until a message is transmitted successfully. when the probability of success on any one attempt is p. 1 2 3 4 5 6,Continuous We begin with the discrete cases. The time in seconds until a lightbulb burns out,t t 0 where is the set of all real numbers.
Applied Statistics Probability 1 10, Definition An event is a collection of points from the. sample space Example the result of one throw of die is. We use sets to describe events, From the die example let the set of even outcomes be E 2 4 6. Let the set of odd outcomes be O 1 3 5, If is finite or countable then a simple event is an event. that contains only one point from the sample space. For the die example the simple events are S1 1 S 2 2 S6 6. Suppose we toss a coin until first Head appears What are. the simple events, Unless stated otherwise ALL SUBSETS of a sample space. are included as possible events Generally we will not be. interested in most of these and many events will have. probability zero,Applied Statistics Probability 1 11.
Describe the sample space and events,Each of 3 machine parts is classified as. either above or below spec,At least one part is below spec. An order for an automobile can specify,either an automatic or standard. transmission premium or standard stereo,V6 or V8 engine leather or cloth interior. and colors red blue black green white,Orders have premium stereo leather interior.
and a V8 engine,Applied Statistics Probability 1 12. Describe sample space and events,The number of hours of normal use of a lightbulb. Lightbulbs that last between 1500 and 1800 hours,The individual weights of automobiles crossing a. bridge measured in tons to nearest hundredth of a,Autos crossing that weigh more than 3 000 pounds. A message is transmitted repeatedly until,transmission is successful.
Those messages transmitted 3 or fewer times,Applied Statistics Probability 1 13. Operations on Events, Because the sample space is a set and any event is a subset A we. form new events from existing events by using the usual set theory operations. A B Both A and B occur,A B At least one of A or B occurs. A A does not occur,S A S A S occurs and A does not occur. the empty set a set that contains no elements,A B A and B are mutually exclusive.
A B Every element of A is an element of B or if A occurs B occurs. Review Venn diagrams in text,Applied Statistics Probability 1 14. Four bits are transmitted over a digital communications channel Each bit is. either distorted or received without distortion Let Ai denote the event that. the ith bit is distorted i 1 2 3 4,a Describe the sample space. b What is the event A1,c What is the event A1 A2,c What is the event A1 A2. d What is the event A1,Applied Statistics Probability 1 15.

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