## Counting Techniques and Introduction to Probability

CountingTechniques and Introduction to Probability

CountingTechniques and Introduction to Probability

Permutationsand Combinations

Atable has been provided which contains information on the number ofmurders by use of different weapons in different states in the year2011 from the FBI’s Criminal Justice Service Division. The firstproblem has two main parts Identifying the number of ways ofselecting five states from the top ten states with most murder caseswhile considering order and doing the same without considering order.This is a typical combination and permutations problem. The firststep is to identify the top ten states with most murder cases. Inreference to the table, the top ten cities in murder cases are,California, Texas, New York, Pennsylvania, Michigan, Georgia, NorthCalifornia, Ohio, Louisiana and Illinois in that order.

Choosingfive states from these ten states while considering order applies therule of permutation. In determining the number of ways five statescan be arranged from these ten states applies this permutationformula:.n represents the total number of times from which the selection isdone. r represents the number of items selected. represents the number of permutations of r items selected from agroup of n items. In our case, n=10 and r=5. Therefore, five cities can be selected in 30240 ways from the top tencities in murder cases when considering order.

Thesecond part involves selecting five states from the top ten states inmurder cases without considering the order in which these states areselected. This is a typical example of a combination problem. Incombination, the order in which items are selected does not matter.The formula for a combination problem is: .n represents the number of items from which the selection is done andr represents the number of items selected. represents the number of ways of selecting a subset of r items from aset of n items without considering the order. In our case, n=10 andr=5. Therefore, five states can be selected in 252 ways from the top tenstates in murder cases when order of selection is not considered.

JointProbability

Thesecond problem involves calculating the joint probability ofselecting California, Texas and New York in that order from a set oftop six states in murder cases. The top six states in murder casesare California, Texas, New York, Pennsylvania, Michigan and Georgia.The probability of selecting the top three states in murder casesfrom the set of top six states in murder cases is a problem of jointprobability. Let the event of choosing the three states be denoted byS and the event of selecting the top three states in a descendingorder be denoted by M. The probability of the two events occurring isgiven by.The probability of event S occurring is also a case of jointprobability. Letting A, B and C represent the events of selectingCalifornia, Texas and New York respectively from the set of top sixstates in murder cases, the joint probability is given by and.Therefore, To find the probability of event M occurring, the number ofpermutations in selecting three states from a set of top six statesin murder cases is determined while putting the order intoconsideration. .Since a subset of three states can be selected in 120 ways from a setof six states, the probability of picking the top three states inmurder cases in a descending order from a set of top six states inmurder crime is given by .Therefore, , which is the probability of choosing the top three states in murdercases in descending order from a set of top six states in murdercases.

Measuresof Central Tendency

Thetop 20 total murders are 1790, 1089, 774, 636, 613, 522, 489, 488,485, 452, 398, 379, 373, 364, 339, 319, 303, 284, 204 and 187. Themean of this data is given by summing this data and dividing thetotal with 20. .is the ithvalue in the array. In this case, .Therefore, .The median of an even set of numbers is calculated by adding the twonumbers in the middle of the array and dividing the sum by 2. In thiscase, .The mode is given by the most occurring number in the sequence. Inthis case, there is no mode.

Measuresof Dispersion

Givena set of n numbers, the variance of the set is given by .Inthis case, .

 1790 1265.6 1.60174e+06 1089 564.6 318773 774 249.6 62300.2 636 111.6 12454.6 613 88.6 7849.96 522 -2.4 5.76 489 -35.4 1253.16 488 -36.4 1324.96

485

-39.4

1552.36

452

-72.4

5241.76

398

-126.4

15976.96

379

-145.4

21141.16

373

-151.4

23839.36

364

-160.4

25728.16

339

-185.4

34373.16

319

-205.4

42189.16

303

-221.4

49017.96

284

-240.4

57792.16

204

-320.4

102656.16

187

-337.4

113838.76

2499052.2

Thestandard deviation is calculated by finding the square root of thevariance.

Probabilities

Theprobability that a person murdered using a firearm in Texas waskilled using a rifle is given by .Let the event of being killed in California using a rifle be denotedR and the event of being killed using a shotgun be denoted by S. Theprobability that the firearm used to kill a specific person inCalifornia was either a rifle or a shotgun is given by .and .Therefore, .The likelihood of a person to be murdered using a handgun can becompared between California and Texas state. The probability of beingkilled using a handgun in California is .The probability of being killed using a handgun in Texas is given by.Therefore, if a person is murdered, the likelihood of being killedusing a handgun in California is higher than in Texas.

Reasonfor the larger number of killings in Texas and California than in NewYork and Pennsylvania in the year 2011

Inmy opinion, the main cause of the disproportionate number of killingsin Texas and California is due to the high population of Hispanics inthese states as compared to New York and Pennsylvania. TheseHispanics commit a lot of crime in these two states including masskillings.