How many bodies were there?

Well, you can either say “5” because that’s all that you have evidence for.

Or you can try to extrapolate based on some figure about how long before the foot pulls away from the ankle. But that would be pure speculation.

How many bodies show up with no feet?

@zenvelo Well, if you said there were a million bodies, then I would say no, if there were a million bodies, there would not be that many pairs in the sample.
And if you said their were 5 bodies, then I would say, it’s highly unlikely that all 5 would be represented in a sample of 7 shoes. It’s more likely that there are six bodies… that kind of argument.

Anyway I’m sure there is enough information here to come up with a best guess. And note, it’s not a forensics problem, but a statistical one.

The total is now 11 feet

@flutherother COOL link! Thank you.

More information is needed to determine the number of bodies, as proven by the following equation:

number of feet found = body count * 2 * probability of finding a foot

We do not know the probability of finding a foot, so we cannot determine the expected number of bodies.

@PhiNotPi You can infer the probability. That is the point of the whole problem. The fact that you don’t like your method is pretty good evidence it’s not the right method, or at least not sufficient reasoning. Had there been NO pairs, I would say any number would be highly speculative. But there are pairs.

Back to what I said before, you know that there are not a million bodies because if you found 7 of two million feet it would be ridiculously unlikely to get two from the same body (remember, all 7 feet washed up alone, and in unique places.)

Put it this way, if I said we all had to bet on how many bodies there were, do you think the best probability expert would come up with a better answer than 47? I do.

Ok, I’ve thought this through some more, and we may be able to get a least a guess as to the number of bodies.

We can use the probability distribution of the probability to find the probability distribution of the number of bodies. (I hope you can understand what I mean). The means that if we believe that there is a 25% chance of there being a 50% chance of any particular foot being found, then we can then believe that there is a 25% chance of being about 11 people.

So, for this analysis, I am going to use the information of 11 feet, of which 2 pairs are from the same people.

This means that there are at least 9 people, who have at least 18 feet. From this, we can conclude that there are at least 7 feet that have not been found. This gives a recovery rate of at most 11/18, or about 61%. If it were higher, then more feet would be found.

However, I haven’t quite found a use for this information.

Your question is not about “probability”, but it asks us – given no more than the snippets of information you have provided (with glaring omissions of “time” and “scope”) – for an estimate of “how many bodies?”

The missing time context means that we don’t know if the shoes / feet discovered were “since the founding of Vancouver” or “since January of this year”, to name a couple of extremes. The missing scope means that we don’t know if these are bodies that may have been disposed of around Vancouver, or anywhere in the Pacific Ocean. We also don’t know how wide-ranging the search for body parts is: Shores of Vancouver Island only? The entire coast of British Columbia? The entire west coast of North America?

In addition, since the shoes and feet that were found were related to “athletic shoes” or sneakers, we would have to make some kind of estimation of how many bodies were unshod, meaning the sharks and crabs have disposed of all but skeletal remains, or were encased in footwear which overcame the bouyancy of any contents, and simply sank out of sight.

You have evidence of five bodies. Because of the sequencing of the paired remains, it is likely that both of those bodies were disposed of nearly simultaneously. The timing of the discoveries is also unknown to us. What was the time between each of the discoveries?

There is simply no way to “estimate” how many more bodies may be involved – from this information. It may be more possible to make such an estimation from “missing persons” statistics, boat traffic in the area up-current from where the remains were found, marine fuel consumption information that may be available, and weather information for the period of inquiry, but not from the data given.

Ok everyone, I’m going to take this out of the realm of all practicality…

Assume that the chance of finding any particular foot is independent of and equal to that of any other foot. In reality, this is not true because objects that are close together will tend to drift together.

Call the probability of find a foot as P and the number of bodies as X. This means that (1 – P) is the chance of not finding a particular foot.

There is a (1-P)*(1-P) chance of a person with both feet that are never found. There are two (1-P)‘s because a person has two feet.
There is a P*(1-P) chance of a person with only his left foot found.
There is a (1-P)*P chance of a person with only his right foot found..
There is a P*P chance of a person with both feet found.

This makes the chance of a person having exactly one foot found is

P * (1-P) + (1-P) * P = 2 * P * (1-P) = 2 * (P – P^2) = 2P – 2P^2

This means that about X*P^2 people have both feet found, while X*(2P-2P^2) people have only one foot found.

Again using the data of 11 feet containing two pairs. There must be at least nine people, two of which have had both feet found and seven of which have only had one foot found.

X*P^2 = 2
X*(2P-2P^2) = 7

2*X*P^2 = 4
X*2*P – X*2*P^2 = 7

X*2*P = 11

(X*2*P – X*2*P^2) / (2*X*P^2) = 7/4
(P – P^2) / (P^2) = 7/4
1/P – 1 = 7/4
1 – P = 7/4*P
1 = 11/4*P
P = 4/11

X*2*P = 11

X*8/11 = 11
X = 121 / 8

So, there are about 121/8 or 15 bodies. We have at least one foot from nine of them. That means that there are 30–11=19 more feet that have not been found.

@PhiNotPi ,that seems like a reasonable answer. I’ll have to look at it some more. I’m troubled that it’s a direct solution rather than something that yields a probability distribution, but who am I to say?

My original problem used different numbers – 7 feet, 2 of them paired. I don’t know if there are still only two paired feet now that the number is up to 11.

As no two feet were sound at the same time or in the same vicinity, it’s reasonable to assume independence of events (though hard to prove!)

@Cwotus, It’s easy to make up extraneous issues. Easier than solving the problem always. Kudos to @PhiNotPi for actually using his noggin.

@flutherother I see your 11 feet and raise you 3 feet.

According to this Wikipedia entry, it is up to 14 feet at this point. not including the one that was a apparently a hoax.

There’s also some interesting information in the entry about foot flotation and drift.

And I really don’t want to speculate on how many bodies, mathematically, are likely to have produced all these paired or single feet. The math is too hard.

If you watch enough episodes of Criminal Minds, the answer will most likely reveal itself. Or you could ask the math genius of the bunch, Spencer Reed. :)

@6rant6 The general solution that I have found is (t^2)/(4n), where t is the total number of feet found, and n is the number of pairs. I also got a friend of mine to verify some of my math.

For fourteen feet (and by reading the wiki page, it seems like there are still two pairs), then the estimate would be 14^2/8, or about 24.5 bodies.

However, one of those findings was that of a human foot and leg bone in a black plastic bag, so it doesn’t really fall into the same category as the others. If we only consider there to 13 feet, then the best estimate would be 13^2/8 or about 21 bodies.

Since the numbers that we are dealing with are relatively small, that brings in a very large uncertainty as to the actual answer. While we can determine that any specific number is the most likely number of bodies, it is also quite possible for that number to be an over- or underestimation. This also means that finding only a few more feet can drastically change the estimate.