The Probabilities of Graduation and Publication

Introduction 

Probability refers to a significant value indicating chances of an event occurrence based on experimental data. Data used to determine the level of uncertainty for an event can be qualitative or quantitative. Probability is, therefore, a qualitative method used by mathematicians to express the possibility of observation or associated risks. Probability derives percentage chance for the event of interest to occur. 

Probability is subjective since the value depends on the data provided or collected from an experiment. The probability value calculated varies when sample space changes (Sahu, 2018). The report entails calculating probabilities of events from the data collected in three universities. The overall probabilities are further used to evaluate instructors' performance based on the number of students graduating and getting publications on the journals. 

Literature Review 

For probability to be applied in prediction, research or experiment must be conducted to collect data. The data collected can be qualitative or quantitative. Each unique observation made is known as an element. The likelihood of an essential element to occur is the probability, and it is known as the sample point for that element. All sample points derived from an experiment form the sample space. When a dice is tossed probability of each value to come at top is the sample point which is 1/6. There are six equal sample points which form the sample space in a case of rolling the dice (Sahu, 2018). 

In a situation where the probability of an event depends on two or more elements of simple games, the situation is referred to as compound events. In our case study probability of student thesis to be published depend on two simple events hence compound because a student must graduate and getting a chance of publication. Binomial distribution has complimentary events where the occurrence of a single event has a probability that describes the other possible outcome. Sample space at any given condition should add up to one or 100% (Glasgow Caledonian University, nd). 

Compound Events 

The two types of compound categories events are mutually exclusive and conditioned events. Mutually exclusive imply events cannot occur simultaneously. If an event precedes the other and has no impact on the probability of consecutive outcomes, they are said to be mutually exclusive. The probability of mutually exclusive compound events is obtained by the addition of simple events of each element. For example, if there are two events A & B, their probabilities are denoted as P (A) and P(B), respectively (Glasgow Caledonian University, nd). 

In a mutually exclusive compound event situation, the probability will be calculated as P (A) plus P(B) 

P (A or B) = P (A) + P (B) 

If events are mutually exclusive, probability of one of the event occurring will be calculated as; 

P (A or B) = P (A) + P (B) – [P (A)* P (B)] 

Conditioned probability is calculated when the outcome of the preceding event has an impact on consecutive events. For example, if there are 3 red and 2 green balls in an opaque bag and two balls are selected randomly without replacement, the sample size for second probability depends is four while that of the first pick is five. The outcome of the other event affects the outcome of the second outcome, and thus the simple point of such an event is said to be conditioned. The probability of an event that depends on precedence is denoted as P (BIA). P (BIA), in simple terms, means the probability of 'B' depends on condition 'A.' Probability of a conditioned mutually exclusive event is presented as (Sahu, 2018); 

P (A and B) = P (A) X P (BIA) 

Data Analysis 

Sample points and conditioned probabilities were computed using inbuilt excel formula on the following conditions. 

The overall probability of students graduating at each university 

P(Graduated) = 

The overall probability of students having a publication at each of the three universities 

P(Publication) = 

The overall probability of students having a publication, given that they graduated at each of the three universities 

P (PIG) = (overall Probability Graduated)* (overall probability publication) 

The probability of a student graduating for each professor 

P(G) = 

The probability of a student having a publication for each professor 

P (P) = 

The probability of a student having a publication given that they graduated for each professor 

P (P) = 

Data Analysis 

P(Graduated)	Rank by P(G)	P(Publications)	Rank by P(P)	P(P|G)	Rank by P(P|G)	Sum of Ranks	Overall Rank
0.78988	0.07855	0.36350	0.09495	0.36350	0.07736	0.25087	0.08362
0.89020	0.04988	0.33848	0.04981	0.33848	0.07204	0.17173	0.05724
0.88017	0.08552	0.31711	0.08093	0.31711	0.06749	0.23394	0.07798
0.95943	0.02044	0.47971	0.02684	0.47971	0.10210	0.14938	0.04979
1.00000	0.10844	0.36006	0.10256	0.36006	0.07663	0.28764	0.09588
0.89938	0.02954	0.31424	0.02711	0.31424	0.06688	0.12353	0.04118
0.93981	0.04128	0.25347	0.02925	0.25347	0.05395	0.12448	0.04149
0.75977	0.04052	0.26597	0.03726	0.26597	0.05661	0.13439	0.04480
1.00000	0.05282	0.48989	0.06798	0.48989	0.10426	0.22506	0.07502
0.84027	0.03798	0.21035	0.02497	0.21035	0.04477	0.10772	0.03591
0.77993	0.11460	0.37439	0.14450	0.37439	0.07968	0.33878	0.11293
0.95005	0.14119	0.29456	0.11498	0.29456	0.06269	0.31886	0.10629
0.78982	0.07967	0.32359	0.08574	0.32359	0.06887	0.23428	0.07809
0.86982	0.11958	0.31324	0.11311	0.31324	0.06667	0.29936	0.09979
	1	4.698561088	1	4.70	1	3	1
P(Graduated)	Rank by P(G)	P(Publications)	Rank by P(P)	P(P|G)	Rank by P(P|G)	Sum of Ranks	Overall Rank
0.76113	0.01384	0.33603	0.01384	0.33603	0.07735	0.10504	0.03501
0.85993	0.10490	0.24072	0.10490	0.24072	0.05541	0.26522	0.08841
0.84004	0.10924	0.36128	0.10924	0.36128	0.08317	0.30165	0.10055
0.88023	0.06488	0.36977	0.06488	0.36977	0.08512	0.21487	0.07162
0.81969	0.06604	0.31181	0.06604	0.31181	0.07178	0.20387	0.06796
0.83007	0.10891	0.29048	0.10891	0.29048	0.06687	0.28468	0.09489
0.79013	0.14426	0.37124	0.14426	0.37124	0.08546	0.37399	0.12466
0.96853	0.01751	0.36713	0.01751	0.36713	0.08451	0.11954	0.03985
0.78961	0.03469	0.27696	0.03469	0.27696	0.06376	0.13314	0.04438
0.79739	0.00901	0.35294	0.00901	0.35294	0.08125	0.09926	0.03309
0.83333	0.00667	0.31746	0.00667	0.31746	0.07308	0.08642	0.02881
0.86016	0.16928	0.40438	0.16928	0.40438	0.09309	0.43165	0.14388
0.79992	0.15077	0.34386	0.15077	0.34386	0.07916	0.38069	0.12690
10.83	1	4.34	1	4.34	1	3	1
P(Graduated)	Rank by P(G)	P(Publications)	Rank by P(P)	P(P|G)	Rank by P(P|G)	Sum of Ranks	Overall Rank
0.81042	0.03498	0.20288	0.02361	0.20288	0.05024	0.10883	0.03628
0.93983	0.12782	0.47009	0.17234	0.47009	0.11642	0.41658	0.13886
0.83985	0.10691	0.22669	0.07779	0.22669	0.05614	0.24084	0.08028
0.85010	0.10557	0.36570	0.12242	0.36570	0.09057	0.31856	0.10619
0.96008	0.11395	0.26895	0.08604	0.26895	0.06661	0.26659	0.08886
0.86019	0.03092	0.27563	0.02670	0.27563	0.06826	0.12588	0.04196
0.85024	0.06059	0.39087	0.07508	0.39087	0.09680	0.23246	0.07749
0.81983	0.02374	0.27107	0.02116	0.27107	0.06713	0.11202	0.03734
0.92727	0.00732	0.28485	0.00606	0.28485	0.07054	0.08393	0.02798
0.75000	0.09289	0.26236	0.08759	0.26236	0.06497	0.24545	0.08182
0.85991	0.12309	0.30090	0.11610	0.30090	0.07452	0.31370	0.10457
0.92976	0.06398	0.35327	0.06553	0.35327	0.08749	0.21700	0.07233
0.88985	0.10825	0.36467	0.11958	0.36467	0.09031	0.31814	0.10605
11.29	1	4.04	1	4.04	1	3	1

Table 1.0: computed probabilities from the data 

Findings 

University	Instructor	Overall Rank	instructor overall %
WWCC	J.W. Blake	0.083621794	8.36%
	K.R. Cunningham	0.057242197	5.72%
	R.H. Doughty	0.07797865	7.80%
	L.M. Edwards	0.049793061	4.98%
	W.H. Greiner	0.095880013	9.59%
	I.D. Jackson	0.041176426	4.12%
	O.P. Lawson	0.041492269	4.15%
	G.F. Nelson	0.04479547	4.48%
	A.F. Paul	0.075021461	7.50%
	D.K. Raulson	0.035906862	3.59%
	T.R. South	0.112925743	11.29%
	E.A. Thomas	0.106287409	10.63%
	C.F. Viney	0.078091813	7.81%
	F.E. Yousef	0.099786831	9.98%
	Totals	1	100.00%
		Overall Rank
EWCC	A.D. Blaise	0.03501309	3.50%
	I.A. Frank	0.088406599	8.84%
	S.D. Gundel	0.100548337	10.05%
	P.O. Hogan	0.071624697	7.16%
	W.M. Kraft	0.067955494	6.80%
	L.I. Luebbers	0.094893343	9.49%
	J.H. Nye	0.124661878	12.47%
	J.A. O'Dell	0.039845592	3.98%
	R.W. Pauly	0.044378763	4.44%
	K.G. Ross	0.033086177	3.31%
	D.S. Smith	0.028807037	2.88%
	J.P. Trost	0.143882423	14.39%
	M.M. Wall	0.12689657	12.69%
	Totals	1	100.00%
		Overall Rank
NWCC	D.H. Allen	0.036277831	3.63%
	T.G. Black	0.138861477	13.89%
	M.A. Carter	0.080279023	8.03%
	M.P. Drake	0.106185665	10.62%
	J.K. Elmsworth	0.088864505	8.89%
	P.T. Grey	0.041959422	4.20%
	C.R. Heines	0.077487111	7.75%
	D.R. Jones	0.037341362	3.73%
	B.M. Keith	0.027975915	2.80%
	G.H. Matheson	0.081817869	8.18%
	P.R. Neighbors	0.104567826	10.46%
	S.T. Orion	0.072334017	7.23%
	A.P. Tracey	0.106047976	10.60%
		1	100.00%

Table 1.1: percentage score for instructors per university 

The final average percentage score for each of the instructor shows their performance ranking in the institution. 

References 

Glasgow Caledonian University. (nd). Probability and Probability Distributions [pdf]. Retrieved from: https://www.gcu.ac.uk/media/gcalwebv2/ebe/ldc/mathsmaterial/level3compnet/Level_3_ Comp_PROBABILITY.pdf 

Sahu, S. (2018). MATH1024: Introduction to Probability and Statistics [pdf]. Retrieved from: http://www.soton.ac.uk/~sks/teach/2018_math1024.pdf