Or was there foul play?

Deaths of people connected in some way—even if it is only reporting on it—to the Madhya Pradesh Professional Examination Board (Vyapam) case have become a regular occurrence, with at least five people associated with the scam having died since 28 June.

Interestingly, most such deaths have occurred seemingly of natural causes that individually would not raise suspicions of foul play.

On Monday, sub-inspector Anamika Kushwaha (recruited through Vyapam) was found dead in a lake. On Sunday, Arun Kumar, dean of N.S. Medical College in Jabalpur, was found dead in a Delhi hotel. Singh died on Saturday.

Other causes of recent deaths include liver failures, and the occasional alleged suicide.

While individual deaths may not arouse suspicion, collectively, foul play cannot be ruled out.

What are the odds of these deaths happening at random?

The pattern of deaths means we will have to do this analysis at different time scales. The first death of a person related to the scam was reported in 2009, when a middleman died of illness and adverse drug reaction.

In 2010, there was a handful of deaths of related people, mostly in road accidents. Since then, there have been a few deaths each year, but the numbers have exploded this year—at least eight people have died, five of them in the last nine days. We will do our calculations at the five-year and nine-day time scales.

What makes the analysis tricky is that the exact number of deaths of people related to the scam is not known. Official sources put the number of deaths somewhere in the mid-20s, while unofficial sources put the body count somewhere in the 40s.

The leader of opposition in Madhya Pradesh has claimed that there have been over 150 deaths related to the scam so far (bit.ly/1KKJjAk).

The other source of uncertainty is the number of people linked to the scam. The Wire reports that at least 1,800 people related to the scam are now in police custody, with another 600 on the run.

This number, however, doesn’t present the full picture of the people associated with the scam, since not everyone associated is either in custody or on the run. Hence, we will assume (note that this is a big assumption) that the total number of people associated with the scam is around three times that number, at around 7,000 (given the scale of approximation, any further precision in this number is unwarranted).

We can assume this number includes beneficiaries, victims, middlemen, whistleblowers, witnesses, journalists, etc.

The way we calculate the odds of the deaths being natural is by asking the question: “Out of a given set of 7,000 Indians, what is the probability that at least 30/40/50 people die in five years?"

In order to calculate this quantity, we will need to use the binomial formula, and one of the inputs it takes is the probability of one of the accused dying in a given year (the probability of dying in five years can be derived from this).

There are different ways to compute this death rate. One option is to use India’s crude death rate (about eight in every 1,000 Indians die every year) as an approximation of the death rate. Another is to determine the age group of the people (The Wire claims that most of the dead are in the 25-30 age group) and use actuarial tables.

In 2013, the Insurance Regulatory and Development Authority (Irda) published the Indian assured lives mortality (IALM) tables based on deaths among holders of life insurance policies in 2006-08 (bit.ly/1S3adUH). A point to be noted is that there is selection bias among people who buy life insurance, with people belonging to upper socio-economic classes being more likely to have life insurance. The same classes have a death rate lower than that of other classes, so this table might represent an underestimate.

Yet, considering that most of the people involved in the scam are people who were either willing to pay for professional seats or jobs, or middlemen who conducted such transactions, we will assume that the Irda numbers are representative of this population and don’t need any adjustments.

A section of the IALM tables (relevant to the age of most victims) is shown in Table 2. We see that the annual death rate in this age group is between 0.1% and 0.11%. This is an order of magnitude away from India’s crude death rate, which is about 0.8% (eight in 1,000), since the latter includes deaths of people of all ages, including the old, who are more likely to die.

Given this disparity, we will go with the actuarial tables and assume that the expected annual death rate of a scam accused is 0.1%. Since this is a small number, the probability of death of a person in this age range in a block of five consecutive years is 0.5%.

So, if we have a pool of 7,000 accused, and the probability of death over a five-year period is 0.5%, we would expect about 35 (0.5% of 7,000) people to have died in this time period, a number that is not far from the actual number of deaths.

The data fits so well that I might be accused of pulling the 7,000 number out of thin air just so that it gives a nice solution. For that purpose, Table 1 shows the probabilities of different death tolls under various assumptions of the total number of people accused in the scam. Each cell in this table is to be read as the probability of at least a certain number of deaths (column header) given that a certain number of people (row header) are linked to the case.

As can be seen from this table, the analysis is highly sensitive to both the number of deaths and the number of people associated with the scam.

For example, for our assumption of 7,000 people having been linked to the scam, the chances that at least 30 people have died naturally is pretty high, but the probability that 40 or more people have died is rather low.

That government sources have tried to keep the number of official deaths low, and that the leader of opposition is quoting an extremely high number are both rational reactions, for a low number of deaths is easier to explain to natural causes.

Yet, this doesn’t tell us the full story. What has brought the Vyapam scam into attention is that at least five people associated with the scam have died in the last nine days. Can we calculate this probability, again using the same assumptions otherwise?

For this purpose, the probability of dying in a year needs to be converted to the probability of dying in nine days. Since we are dealing with small numbers, we can simply make a linear interpolation, so if the probability of dying in a year is 0.1%, probability of dying in a period of nine days is 0.1% x 9 / 365 = 0.0025%.

From a pool of 7,000 given Indians in a 25-30 age group, we would expect 0.0025%, or 0.17, people to die in a period of nine days! The actual number of deaths (5) is an order of magnitude larger. The probability that at least 5 of the 7,000 people die in the space of nine days is one in a million!

While the probability of 5 out of 7,000 people dying in a given block of nine days is one in a million, this is not precisely what we should be looking for, because since the beginning of the scam, there have been several blocks of nine days. What we should be looking for is the probability of five people dying in any block of nine days since the breakout of the scam. And for his purpose, the above number must be multiplied by the number of nine-day blocks. If we consider a period of 10 years after the breakout of the scam, there are about 3,650 nine-day blocks in that period (day 1-9, 2-10, 3-11 and so on).

If the probability of five deaths in a given block of nine days is one in a million, the probability of five deaths in any block of nine days over 10 years is one in a million times 3,650, which is 0.365%. Significantly higher than the previous number, but still low enough for us to dismiss the chances of it being a random occurrence.

The overall number of deaths in the last five years, if it is about 40 as most sources claim, does not raise much concern given the large number of people associated with the scam. Yet, if we look closely at the deaths in the past few days, the chances of foul play are indeed extremely high.

Subscribe to Mint Newsletters * Enter a valid email * Thank you for subscribing to our newsletter.

Share Via