Mathematical Proof of Conspiracy Against Bill Cosby

Time Magazine Cover

This is the outline for a deeper scientific study that needs to be done.

Saturday Evening Post

Bill Cosby is one of the most important civil rights leaders, educators, and entertainers in the history of the United States. In 2005, when he was 65 years old, five women publicly accused him of drugging and sexually assaulting them. One claimed it happened one year in the past [1], two claimed it happened about 20 years in the past[2] and two claimed it happened 30-35 years in the past [3]. Nobody accused him for the next nine years. In 2014-2015, when he was 77 years old, some 35 more women accused him on television and in the mass media of drugging and assaults, from 7-50 years in the past,  All of the public accusers in 2005 were part of a lawsuit against Cosby filed by attorneys Deloris Troiani and Bebe Kivitz.  Apparently six of the 2014-15 accusers were non-public accusers from the 2005 lawsuit.

Most of the new accusers in 2014-15 were clients of attorney Gloria Allred or her associates. Besides the 40 women accusing him of drugging and sexual assault, an additional 20 women went to the mass media and made other much less sensational accusations against him (ranging from “rough kissing” to “spreading rumors” to “touching a woman’s breast in a public restaurant”).

All of the accusers, with the exception of the first, admitted to seeing or hearing reports of Cosby drugging and assaulting women in the mass media before going public with their claims. This suggests that the mass media coverage was an important factor in their making these claims.

The one thing that all the drug and sexual assault accusers have in common is that at the time of the incident/incidents, none went to the police to file a report, or to a doctor, clinic or hospital to be tested after being drugged, none got a lawyer to file a lawsuit, and none went to a reporter to offer their stories. None wrote a letter or made a diary entry. None told a relative or a friend who wrote a letter or publicized it in any way.

What are the odds that all these woman are telling the truth about the drugging and sexual assault and none of them going to the police, or a doctor, or a clinic, or a lawyer, or talking to a reporter or writing a letter, (either signed or anonymous), or telling relatives or friends who did any of these things?  This article attempts to quantify those odds. While no scientific evidence exists for calculating the odds of an accuser not doing all of the these things, there is scientific evidence available to calculate the odds of an accuser not going to the police. Logically, not going to police does not preclude women from going to a doctor, clinic, lawyer, reporter, or writing a signed or anonymous letter about it, or telling relatives or friends who did any of these things. Thus we would expect women to be able to present more physical evidence than just going to the police and generating police reports.

The idea that one of the most well known, beloved, celebrated, and honored men in American history should also be a secret serial criminal suggests a fantastic breakdown in the American justice system for 50 years or a mass conspiracy against Dr. Cosby. For the purposes of this article, we shall define a conspiracy according to 18 U.S.C. 371: which makes it

a separate Federal crime or offense for anyone to conspire or agree with someone else to do something which, if actually carried out, would amount to another Federal crime or offense. So, under this law, a ‘conspiracy’ is an agreement or a kind of ‘partnership’ in criminal purposes in which each member becomes the agent or partner of every other member. as any untrue accusation made public with the help of lawyers and the mass media[4].

Reasons for Not Going to Police that are Not Scientifically Backed

Accusers and others have given numerous reasons for not reporting the crimes in a timely fashion. For example, that the perpetrator was a wealthy, black man. There is no scientific evidence that victims of rape are less reluctant to go to the police when their attackers are wealthy or black than when their attackers are poor or white.

Frequently, the accusers say that they feared retaliation in their careers. There is evidence that sexual assault victims are more reluctant to go to the police when they fear physical or other retaliation. However, Dr. Cosby was a comedian and had never shown any significant indication of violent behavior, there is no reason to believe that any of the accusers felt physically threatened by Cosby. Because of his extreme gentleness and warmth, demonstrated in thousands of live concerts, hundreds of television shows, hundreds of well publicized charitable activities, and testimonies from hundreds of people who worked with Cosby, none of the accusers would have had any reason to expect any physical retaliation for going to the police. As far as retaliation to their careers, since Cosby was only one of hundreds of Hollywood producers, there was also no reasonable expectation that he could hurt their careers if they went to the police. One can question if there is any evidence of Hollywood or New York actors or actresses being blacklisted or seriously hindered from work over the last 50 years due to criminal reports or lawsuits. Other explanations, such as Cosby being too powerful to report to the police, lack any scientific evidence to back them up.

In fact, models and actresses, the vast majority of the accusers, are taught to act brash and bold, and often welcome publicity.  One can logically conclude that they are far more likely to go to the police than the average person

Reported and Unreported Rape Statistics
While not all rapes and attempted rapes are reported to the police, a great number are. Since 1973, the Department of Justice has been using annual surveys to scientifically determine how many rapes/attempted rapes are reported and go unreported. Based on these reports we can determine mathematically the odds that all these women were actually assaulted and did not go to the police.

By adding up the percentage of reported rapes each year from 1973-1992, and dividing by 20, we find that the average annual number of reported rapes was 52%. This means 48% went unreported annually on average.  (Note: all data can be found at It is in “Criminal Victimization 1992”).

Criminal Victimizations 1992 Table 5 Reporting Victimizations to the Police
Total = 1044% / 20 years = 52.2% reported per year on average

Rape was defined this way in the surveys from 1973 to 1992. (Note: from “Crime Victimization in the United States, 1991.”)

Rape — Carnal Knowledge through the use of force or the threat of force, including attempts. Statutory rape (without force) is excluded. Both heterosexual and homosexual rape are included.

The Important Change in the Definition of Rape in 1993 in the Annual Crime Surveys
In the 1993 survey published in 1996, the definition of rape was drastically broadened. (Note: from “Crime Victimization in the United States, 1993”)

Rape — Forced sexual intercourse including both psychological coercion as well as physical force. Forced sexual intercourse means vaginal, anal or oral penetration by the offender(s). This category also includes incidents where the penetration is from a foreign object such as a bottle. Includes attempted rapes, male as well as female victims, and both heterosexual and homosexual rape. Attempted rape includes verbal threats of rape.

The main and essential difference is that “psychological coercion” has been added. The term “psychological coercion” is extremely broad and can be interpreted in any number of ways. It is often impossible to distinguish between “coercion” and “offers.” For example, if a man offers a woman cocaine, or a mink coat, in exchange for sex, that can be considered coercion and therefore rape. If a woman offers a man sex in exchange for cocaine, or a mink coat, that is considered an offer and not considered rape. Yet the actions will be exactly the same in both situations.
The effect of broadening the definition of rape to include non-violent “psychological coercion” was to cause a jump in the unreported rape rate. In 1993, unreported rapes jumped from an average of 48% that it had been for the previous 20 years, to 71%. This was 12% higher than the highest previous one year number of unreported rapes (59% in 1980) in the 20 year period from 1973-1992. Over the next 18 years, from 1993 to 2010, the average number of unreported rapes was 66%.
Victimizations not reported to the Police 2006 to 2010
There is no evidence and no reason to believe that the police reports of violent rape/rape attempts went down from 1993 to 2010. Rather it is probable that the expansion of the definition to “psychological coercion” caused the increase in unreported rapes from 48% average per year to 66% on average.

None of the drug and sexual assault accusations against Dr. Cosby would have been reports for non-violent psychological coercion. Because they involved drugging, they would have been reports for actual violent rape/rape attempts. Therefore it makes sense to use the 52% reported violent rape/rape attempts that was established from the data from 1973 to 1992.Also, the majority of the incidents are alleged to have happened during this time.

This means that if a drugging and sexual assault did take place, there was a 52% chance of that person going to the police. Cosby would have had a 48% chance of the woman not going to the police. This means that the chances are 52% that any specific one of the accusers would have gone to the police. If  the first two accusers are telling the truth, Cosby would have had a .48 x .48 chance or a .23 chance of not being reported by one of these two women. There is thus a less than a one in four chance that any two of the 40 accusers selected at random are telling the truth.  If the first three women are telling the truth, Cosby would have had a .48 x .48 x .48 chance or an .11 or 11% chance of not getting investigated by the police. Another way of looking at it, is that the odds are 89% that at least one of the first three women are not telling the truth.
Assuming these women were normal assault victims and had a normal chance of reporting to the police, here are the odds of the chances that 1-40 of the women are telling the truth about assault  and no one going to the police:
One accuser accusations true – 48%, accusations false – 52%
Two accusers accusations true – 23%, accusations false – 77%
Three accusers accusations true – 11%, accusations false – 89%
Four accusers accusations true – 5%, accusations false – 95%
Five accusers accusations true – 3%, accusations false – 97
10 accusers accusations true – .065%, accusations false – 99.94%
20 accusers accusations true – .00000042% (42 in 100 million), accusations false – 99.999958%

30 accusers accusations true – .00000000027 (27 in 100 billion), accusations false – 99.999999973%

40 accusers accusations true – .00000000000018 (18 in 100 trillion), accusations false – 99.99999999999982%.

Thus the odds of all of the accusers telling the truth are 18 in 100 trillion or less than 2 in 10 trillion.

Calculation of Confidence Level

Let us assume for our null hypothesis that The women accusing Dr. Cosby are not a random and independent sample from the population in the study of women rape victims from 1973-1992 as portrayed in the U.S. Department of statistics when it comes to reporting to the police.

Since the number of alleged drug and assaults was established at 52%, we can round that off to 50% or the same odds as a coin flip which makes the probabilities a binomial distribution. We can calculate a confidence level for the 40 allegations that we are dealing with.

For a binominal, Standard deviation = SQR{(N*p*(1-p)}.

Thus, Standard deviation = SQR{(40*.5*(1-5)} = SQR{(40*.5*.5)} = SQR 10 = 3.16227766017 = ~ 3.162

Confidence level for one standard deviation or Z is 68.8%, thus sample average plus or minus sample standard deviation equals  x̅ (±)s. Since the average of people going to the police in a binomial sample of 40 is 40/2 or 20, we can say that 20 (±) 3.162 or between 16.838 – 23.162 would go to the police with a 68.8% confidence level.
A Z score of 2.59 would give us a 99% confidence level. Thus 2.58(Z) = 2.58(3.162) = 8.15796 or ~8.16. The 20 average reports would allow us to expect 20 (±) 8.16 or between 11.84 to 28.16 police reports with a 99% confidence level.

Since 12 to 28 of the alleged victims did not report to the police, we can say with a 99% confidence level that the null hypothesis is wrong. The women accusing Dr. Cosby are not a random and independent sample from the population of women rape victims from 1973-1992 as portrayed in the U.S. Department of statistics when it comes to reporting to the police.

Does this mean that the odds are equally great that some of the accusers are telling the truth?

The odds that an accuser is not telling the truth increases exponentially with each additional accuser. With 40 accusers, we should have expected 20 or 21 to have gone to the police. The fact that none did, can only be reasonably explained by this not being a random sample of sexual assault victims, which points in the direction of a conspiracy. This should not be taken to mean that it is likely that 19 or 20 are telling the truth. If half are implicated in a conspiracy, there is reason to believe that the other half are also part of that conspiracy (or conspiracies). If we find that 40 people claim to see Big Foot or a wizard flying through the air, and we find that 19 of them are lying, it does not mean that the rest are telling the truth, but it suggests strongly that all are lying or mistaken and have entered into a conspiracy with each other.

It should be recognized that going to the police was only one of a number of ways that the accusers could have provided evidence for their claims. They could have had drug tests done, hired lawyers, called reporters, written letters to friends or relatives, written anonymous letters to the mass media, written to companies employing Dr. Cosby, or told friends and relatives, who, in turn, could have notified authorities or written letters.

Based on their not reporting their alleged crimes to the police, and Department of Justice crime statistics, there is at least a 52% chance of any accuser falsely accusing Dr. Cosby.  The lack of any written evidence generated at the time that the incidents were alleged to have occurred would and admissions by all the accusers that they did not seek medical or legal or write any complaints or include direct references to the event increase those odds enormously. The chances that these accusers are telling the truth and are not part of a conspiracy become less and less as more accusers come forward. The chances of this being a conspiracy against Dr. Cosby increase exponentially with each accuser. The chances of conspiracy against Dr. Cosby are at least a trillion to one in favor, unless some other reasonable explanation for the lack of evidence to confirm any of the accounts is found.

Basically, a 52% chance of going to the police is very close to 50%, which is the odds of a coin flipping heads or tails. If all the women are telling the truth, each time that there was a drugging and assault, Cosby was flipping a coin. According to them it came up the same  40 times in a row. This gives us a result of .0000000000009 or 9.094947e-13.The odds against this happening are .5 to the 40th power or 9,999,999,999,999 to 1 (~ten trillion to one). Assuming that the women are a randomly and independently selected group of women, the odds that there is a conspiracy against Dr. Cosby are greater than 10 trillion to one.

We also found with a 99% confidence level that 12-28 of the 40 accusers should have gone to the police if all 40 of these people had been assaulted. The fact that none did suggests that this was not a random and independent sample of actual rape victims.



[1] Andrea Constand

[2] Barbara Bowman, Beth Ferrier

[3] Shawn Berkes, Tamara Green