Probability Coin Toss Experiment


Subjective, Empirical, and Computational

Probability for Coin Toss Experiment


This simulation is designed to demonstrate the difference between subjective, empirical, and computational probabilities. A simulation was used that records the result of flipping a coin ten times concentrating on the number of heads obtained. This simulation was run sixty times and the results were entered into an Excel spreadsheet for analysis. First subjective probabilities were generated from educated guesswork. Then empirical probabilities were generated based on the results of the simulations. Finally computational probabilities were calculated using the binomial distribution formula. Both the subjective and the empirical probabilities were compared with the computational probabilities in order to emphasize the differences.

To begin with there are several questions regarding the subjective probabilities associated with the eleven possible results. Since five heads will occur more frequently than any other combination, it should be significantly greater than the average value of 9.1%. The assigned value for obtaining five heads is 22%. Since the binomial distribution is symmetric the probability of getting three heads will be the same as the probability of getting seven heads. Knowing that the total number of heads must be between zero and 10 and also that the sum of all the percentages must equal 100% exactly these were assigned a subjective probability of 12%. Again since the binomial distribution is symmetric the probability of getting no heads will be the same as getting all heads. The subjective probability assigned to both of these results was 1%.

From this point the remainder of the subjective probability chart was filled out knowing that the distribution was symmetric, each value had to be between 0 and 1, and the sum of all the probabilities had to be exactly 1. After a little bit of experimentation the table at the top of the following page was generated:

X = Heads 0 1 2 3 4 5 6 7 8 9 10
Prob (X) 0.01 0.02 0.05 0.12 0.19 0.22 0.19 0.12 0.05 0.02 0.01

Examination by eye confirms that the distribution is symmetric and each value is within the allowed range. For the final requirement it is readily confirmed that

2 (0.01 + 0.02 + 0.05 + 0.12 + 0.19) + 0.22 = 1.00

and an acceptable subjective probability distribution has been generated.

Assuming that a fair coin is being used, the probability of getting heads on any toss will be 0.50 as will be the probability of getting tails. This means that over an extended period of time half of the tosses are expected to be heads and half of them are expected to be tails. As a result for a total of 600 tosses the expected value for heads is 300. Also since the probabilities of success (heads) and failure (tails) are equal this means that the probability of getting three heads will be exactly equal to the probability of getting three tails. Finally, the probability of getting three heads is exactly equal to the probability of getting seven tails. This is due to the fact that this is a binomial distribution and the only possibilities are heads or tails. If exactly three heads are generated then the other seven tosses must have resulted in tails.

The first graph presented for discussion is a line graph showing the evolution of cumulative heads percentage over all 60 simulations. It would be expected that this cumulative percentage would approach the success probability over a long enough timeline. Since the probability of success in this case (assuming a fair coin) is 0.50 the line graph should approach this value with decreasing fluctuations as the number of simulations increases. This line graph is presented at the top of the following page:

The sharp jump at the beginning simply represents an abundance of heads after a short number of tosses. Around the 25th toss the cumulative probability returned to its expected value and then dipped below 0.50 showing an overall abundance of tails. As expected the line graph continued to fluctuate around 0.50 with the degree of fluctuation decreasing as the total number of simulations increased.

The second graph is a comparison of the subjective probabilities to the computational probabilities calculated. The subjective probabilities can be thought of as a gut instinct while the computational probabilities are exact based on the binomial distribution formula. This graph is presented at the top of the following page:


The subjective probabilities are shown in blue and the computational probabilities are shown in red. For this particular subjective probability distribution the central values were underestimated while the outer values were overestimated. Notice that both distributions are symmetric as would be expected. The overall logic error in the subjective probability distribution appears to be a misunderstanding of how quickly the probability decreases as the number of successes deviates in either direction from the expected value of five. Also notice that the standard deviation of the subjective distribution is larger than that of the computational distribution.

The final graph is a comparison of the empirical probabilities to the computational probabilities. The empirical probabilities are determined from the sixty simulations: empirical probability distributions are expected to approach the computational distribution as the number of trials increases. Again this graph is presented at the top of the following page:


Interestingly enough the empirical distribution also appears to underestimate the computational distribution for the central values and overestimate it for the outer values. Notice here that the empirical distribution is no longer symmetric, and there is no reason to expect that it would be. It appears that sixty simulations are enough for the empirical distribution to be recognizable compared with the computational distribution. In other words the empirical distribution is a reasonable approximation but is not exact.

In conclusion this experiment is very good for showing the differences between the three types of probability distributions. It is simple to execute and the fact that the probabilities of success and failure are equal make the subjective distribution easier to analyze. All three distributions are relatively similar and all of the results were as expected.



The first thing that I learned was that there is actually some reasoning that goes into determining a subjective distribution. Before this experiment I felt it was a mostly useless exercise, especially if the computational distribution was readily available. Now I realize that with a bit of logic it is possible and actually not too difficult to generate a reasonable subjective distribution. The advantage of this is that for situations where the computational distribution is more difficult a ballpark idea can be gathered with relative simplicity.

The next thing that I gained from this project is a better interpretation of the line graph for cumulative percentage of heads. I knew that it would approach the expected value, but I never really thought about what the fluctuations really meant. No I understand that when the cumulative probability is greater than the expected value that there is an abundance of successes while when the cumulative probability is less than the expected value there is an abundance of failures. The last thing that I learned was that a relatively small number of simulations can generate a reasonable empirical distribution. With eleven different possible outcomes I would have thought that considerably more than sixty simulations would have been needed.

On a slightly different note I also learned some new functions for the Excel spreadsheets. The most interesting one to me was the command to generate the exact probabilities for a binomial distribution. All the other functions I used were familiar to me, but it was good practice to use them again. I feel like this project helped me learn and reaffirm quite a bit of knowledge related to probability distributions and the use of Excel to analyze them.



Toss Number Cumulative Total Coins Cumulative
Number Heads Number Heads Tossed Percent Heads
1 5 5 10 0.5000
2 7 12 20 0.6000
3 5 17 30 0.5667
4 6 23 40 0.5750
5 4 27 50 0.5400
6 7 34 60 0.5667
7 3 37 70 0.5286
8 4 41 80 0.5125
9 6 47 90 0.5222
10 8 55 100 0.5500
11 6 61 110 0.5545
12 5 66 120 0.5500
13 6 72 130 0.5538
14 5 77 140 0.5500
15 3 80 150 0.5333
16 3 83 160 0.5188
17 7 90 170 0.5294
18 5 95 180 0.5278
19 6 101 190 0.5316
20 2 103 200 0.5150
21 6 109 210 0.5190
22 5 114 220 0.5182
23 3 117 230 0.5087
24 2 119 240 0.4958
25 6 125 250 0.5000
26 2 127 260 0.4885
27 8 135 270 0.5000
28 4 139 280 0.4964
29 3 142 290 0.4897
30 7 149 300 0.4967
31 5 154 310 0.4968
32 3 157 320 0.4906
33 4 161 330 0.4879
34 6 167 340 0.4912
35 9 176 350 0.5029
36 5 181 360 0.5028
37 7 188 370 0.5081
38 6 194 380 0.5105
39 4 198 390 0.5077
40 2 200 400 0.5000
41 5 205 410 0.5000
42 8 213 420 0.5071
43 4 217 430 0.5047
44 5 222 440 0.5045
45 5 227 450 0.5044
46 6 233 460 0.5065
47 8 241 470 0.5128
48 7 248 480 0.5167
49 4 252 490 0.5143
50 4 256 500 0.5120
51 0 256 510 0.5020
52 4 260 520 0.5000
53 7 267 530 0.5038
54 6 273 540 0.5056
55 5 278 550 0.5055
56 2 280 560 0.5000
57 3 283 570 0.4965
58 5 288 580 0.4966
59 4 292 590 0.4949
60 7 299 600 0.4983


Table 1.

Simulation number, Total number of heads, Cumulative number of heads
Total number of coins tossed, Cumulative percentage of heads



Number Subjective Actual Empirical Computational
Successes Probability Successes Probability Probability
0 0.010 1 0.017 0.001
1 0.020 0 0.000 0.010
2 0.050 5 0.083 0.044
3 0.120 7 0.117 0.117
4 0.190 10 0.167 0.205
5 0.220 13 0.217 0.246
6 0.190 11 0.183 0.205
7 0.120 8 0.133 0.117
8 0.050 4 0.067 0.044
9 0.020 1 0.017 0.010
10 0.010 0 0.000 0.001
Sums 1.000 60.000 1.000 1.000



Table 2.

Number of successes, Subjective probability, Actual successes

Empirical probability, Computational probability


Effects of Trichloroethylene to the Kidneys



Effects of Trichloroethylene to the Kidneys

Toxins hinder the normal functioning of the body by interfering with the mechanism through which the body organs and systems operates. Different toxins affect different body organs including the kidney. This research paper involves a review of an article that discusses the effects of trichloroethylene on kidneys. The article being reviewed is titled “Mechanisms of Toxicant-Induced Acute Kidney Injury”. The article is to be found in a journal called Science Direct. The article examines how the functions of the kidney are impaired by trichloroethylene due to the injuries caused by this toxicant to the kidney.

Trichloroethylene as well as its metabolites has been identified to be toxic to the kidneys as it has been found to trigger the development tumors in the kidneys and the toxin is therefore responsible for the development of kidney cancer (Kacew & Lee 2013). Studies done in animals and humans relating to the effect of trichloroethylene to the kidneys have revealed similar results. In these studies, rats and humans were exposed to trichloroethylene and the damage to both was similar. In both, the proximal tubule was damaged.

Evidence from mechanistic, experimental as well as epidemiologic studies has shown that trichloroethylene has the potential to cause cancer of the kidney. In these studies, male and female rats were exposed to the toxin the results obtained showed that male rats were more at risk of developing tumors as compared to female rats. Studies in humans showed that exposure to trichloroethylene activated the development of cancerous cells (Lash, 2014). Studies conducted on animals showed that trichloroethylene operates as a complete carcinogen. The effect of trichloroethylene is prevalent al all the stages of cancer development starting from tumor formation, promotion as well as progression. The speed at which the development and progression of the disease largely depends on the dosage in terms of concentration as well as the time that one is exposed to the toxicant.

The most common type of cancer that is associated with trichloroethylene is the renal clear cell carcinoma. The toxin stimulates the activation of VHL which helps in suppressing the gene responsible for the formation of tumor. Evidence from studies has shown that there is a very close relationship between trichloroethylene and mutation of the VHL especially in human beings. In certain studies, it has been determined that exposure to high concentrations leads to mutation of the renal cancer cells.

In addition to explaining the mechanism through which trichloroethylene causes renal cancer, the article as well explains how trichloroethylene gets into the human body. Trichloroethylene is among those toxicants that are found within the environment in large quantities owing to their varied uses. Trichloroethylene is used both at home as well as in the industries. In addition, trichloroethylene is also to be found in the metabolites and this increases the chance of exposure. It is estimated that tens of millions of pounds of trichloroethylene are released in the environment each year in the United States alone. These figures clearly indicates that the level of exposure to this chemical is quite high within the population.


Regulation of cosmetic products

The Food and Drug Administration does not control the chemicals that are used in a perfume to enhance its safety contrary to what many people think. Although the law requires the companies that produce cosmetic to indicate in the label the chemicals that have been used, the same law allows companies to keep secret of the chemical formula that have been used to make fragrances (Cheryl, 2010). Cosmetics producing companies can take advantage of this lapse and introduce dangerous chemicals in their products. There should be a body that is mandated to administer the cosmetics industry fully in order to enhance the safety of the consumers. This role should not be given to FDA as cosmetics do not fall under food or drug. A new regulatory body should be created to promote the safety of cosmetics.














Cheryl, S. (2010). Trichloroethylene Health Risks–State of the Science.” Environmental Health

Perspectives Supplements 108(2) 56-72

Kacew, S., & Lee, B. M. (2013). Lu’s basic toxicology: Fundamentals, target organs, and risk

assessment (6th ed.). New York, NY: Informa Healthcare.

Lash, H. (2014). Mechanisms of Toxicant-Induced Acute Kidney Injury, Science Direct Journal,

7(4) 81-115

Information Literacy Skills




Information Literacy Skills


Student Name:

Date of Submission: August 28, 2015


Logo Design

Question 1

Students need to be information literate to enable them learn and gather knowledge effectively. This is because today’s education system is characterized by an information explosion over varied sources including the internet. Information literacy therefore bestows upon them, valuable skills for information, research, use and communication.

Question 2

“Just over 80% of students feel adequately or somewhat prepared to conduct research BUT ONLY 16% feel very prepared to do research.” I find this most surprising following the small number of students who are actually ready to carry out the research despite the huge percentage of the students feeling ready to do so. This probably depicts the factors that underpin research challenges faced by graduates in the workplaces. The most significant statistic is that 97% of students can define plagiarism. This means that in as much as 74% have admitted violations of the academic integrity policy at least once, they remain liable for the offense. Such violations seem to be deliberate. More strict actions can therefore be taken against such students who are aware of plagiarism, but still go against the policy.


There are no sources in the current document.



Homosexuality in Greek Society







In the ancient times, such philosophers as Plato, Herodotus, Athenaeus, and Xenophon explored elements of same-sex relationships in the ancient Greece. This form of sexual relations was evident between mature men and teenage boys, an aspect referred to as pederasty. Despite the existence of homosexuals in this society, some of these men defied the existing social principles by assuming an inactive sexual role. Nonetheless, there were minimal examples of homosexual relationships between females. Unlike the modern western communities, ancient Greeks did not perceive sexual orientation as an identifier in the society. As opposed to using the gender of the involved individuals to distinguish their sexual behaviors and desires, the Greek society focused on the participants’ roles in the sexual act by classifying them into active or passive parties (Osborne 35). The role of a dominant participator was linked to masculinity, adulthood, and high social status. In contrast, the passive sexual role was related to youth, low social status, and femininity.

Homosexuality among Ancient Greek Men

Pederasty, which referred to boy love, was the commonest form of homosexual relations between Greek men.  It involved mature men and teenage boys. In Athens, the mature male was referred to as an erastes and had the responsibility of protecting, educating, loving, and being a role model to his young lover. The reward from his eromenos, the teenage boy, was his promise, youthfulness, and beauty. Pederasty began prior to the establishment of the city-state as one of the divisions within Greece’s political framework. The encompassed tribal districts were classified into age groups. As part of the community’s rite of passage, the young men were placed under the care of older males for a significant period as a way of being enlightened on the acceptable societal principles and adulthood responsibilities (Osborne 39).

This practice evolved into pederasty. The young boys in the Greek society did not leave the community borders but rather intermingled with mature men within the city. These older males had an instructive and educational responsibility in the boys’ lives as part of their sexual relationship. Nonetheless, penetrative sex was considered humiliating for the inactive partner. It was one of the unacceptable societal norms (Neill 99). This is mainly because of the comprehensive social code that governed pederasty in the Greek community. To begin with, it was the responsibility of the older male to court the teenager he fancied although the young man had to hold back for a while before giving in to the sexual advances. This period enabled the young man in question to evaluate the intentions of his suitor. A suitable suitor not only showed his sexual intentions towards his subject but he also had to have genuine affection for him.  Although the age limit of the teenagers involved in the Greek pederasty was 12 years, no legal penalties were attached to the defiance of this social code. Ancient Greeks were the first group to consider pederasty as an educational and social institution (Osborne 50). It was a key aspect in the military, civil life, arts, and philosophy. However, analysts have differing arguments regarding the confinement of pederasty to upper societal crust as opposed to its expansion to different social classes.


the same-sex relationships between soldiers involved the Sacred Band of Thebes is one of the classic examples of pederasty in the ancient Greek society. It was a special military unit that only comprised of men and their lovers. This was perceived as a suitable approach to boost the troop’s fighting spirit. Such a bond is reflected in lliad through the epic relationship between Patroclus and Achilles (Osborne 49). These relations were perceived to enhance one’s bravery and morale owing to the soldiers’ desire to protect and impress their lovers. Similarly, during the Lelantine War, Chalcidians sought the help of Cleomachus before engaging in a battle with the Eretrians. This glorious warrior responded to the request and brought his lover to see the fight. Cleomachus led the Chalcidians to victory against their opponents at the expense of his life (Osborne 52). The society perceived the warrior’s disregard for his life in order to save his lover as the noblest act.

In addition, homosexuality was also evident between two mature men. However, relationships between mature males from similar social status were viewed as problematic. This is mainly because of the essence of masculinity among adult men and the observed feminizing effect among passive partners in the Greek society. Accordingly, same-sex relationships between adult males attracted social stigma. Nonetheless, this disgrace was only directed towards the passive partner in such a relationship. For this reason, Greek men who took the passive role in a sexual relationship even after entering adulthood were feminized. This is because, during this period of a man’s life, the young male was expected to take the active position in the pederastic relationship. Some of the recorded adult male couples in the ancient Greek society include Agathon, who was a poet, and Pausanias from Athens (Neill 81).  Similarly, the relationship between Hephaestion and Alexander the Great has been categorized into this group of relations.

Homosexuality among Ancient Greek Women

The initial occurrence of same-sex relationships between women was evident in Sappho’s texts. This is mainly because she wrote most of her poems to women. At times, the love encompassed in these texts appeared to be a way of seeking revenge. Some analysts identified 12,000 poetry lines that highlighted her affection for other females. Other than being a poet, Sappho was the leader of thiasos. This was a community of ancient Greek females where the members received some form of education (Osborne 126). Some girls in this community were involved in homosexual relations. This female group was suppressed following the portrayal of marriage as a key institution within the ancient Greek culture and the resultant confinement of women to the home setting. Throughout their lives, girls were required to show their love for their husbands. Accordingly, female homosexuality was discouraged in the established Greek society. In addition to females’ athletic nudity, there are certain erotic relationships recorded for Sparta.  Similarly, Plato’s Symposium highlights a group of women who had female attachments and had minimal regard for their husbands (Osborne 85).

Homosexuality in the Modern Greece Society

Despite the legalization of same-sex relationships in Greece, the lesbians, gays, bisexuals, and transgender (LGBT) community faces certain challenges owing to societal stigma. On one hand, since 2006, male prostitution has been a legal practice in Greece. Nonetheless, lesbians are neither recognized nor mentioned in the nation’s Criminal Code. This is evident in Article 347, which stipulates the age limit for consensual sexual activities between partners of the same gender (Hubbard and Verstraete 128). In addition, the legalization of marriage between homosexuals is an attempt by the government to suppress societal discrimination that may be triggered by differences in the citizens’ sexual orientations. Although the government of former Prime Minister Costas Karamanlis was against these relationships, various court cases have shown the legality of same-sex marriages and the essence of suppressing discrimination based on one’s sexual orientation. For instance, in 2013, the European Court Of Human Rights presented a verdict that favored homosexuals in a case between the Greece government and Valianatos and others. In his ruling, the judge expressed his disapproval of the centralized administration’s attempts to exempt homosexual couples from civil unions (Hubbard and Verstraete 130).

Similarly, promotion of the LGBT rights was evident in 2005 when discrimination of homosexuals in the workplace was illegalized (Hubbard and Verstraete 149). This proposed legislation aimed at protecting individuals from discrimination based on their gender identity or sexual orientation. Nonetheless, various religious denominations have highlighted their discontentment on the legalization of same-sex marriages. For instance, the Orthodox Church considers homosexuality as a sin against God and a flaw of humanity (Hubbard and Verstraete 152). In conclusion, there are certain similarities between homosexuality in the ancient and modern Greek societies. Various parties in both communities accept such relationships although there are some individuals who have shown their disapproval of such relations.




Works Cited

Hubbard, Thomas K, and Beert C. Verstraete. Censoring Sex Research: The Debate over Male Intergenerational Relations. , 2013. Print.

Neill, James. The Origins and Role of Same-Sex Relations in Human Societies. Jefferson, N.C: McFarland & Co, 2009. Print.

Osborne, Robin. Studies in Ancient Greek and Roman Society. Cambridge, UK: Cambridge University Press, 2004. Print.