Friday 1 October 2010

The Plausible versus the Probable

This blog occasionally (always?) takes on what is the new trend in Skepticism. It has always been around a long time but currently it is making more noise than usual and it is no surprise that Loch Ness Monsters should come under their cannon fire.

Recently I have been defending the Spicers who claimed in 1933 to have seen Nessie cross the road (to get to the other side no doubt).

If one is a skeptic then it is bad enough to suggest large entities inhabit Loch Ness - but to also forage onto land offends logic in the extreme!

I thought I had laid to rest any notions of otters or deers crossing in front of the Spicers but Alexander Lovcanski put out an interesting piece that the Spicers saw a mirage enhanced otter that day. His analysis is here.

Okay, it deserves to be aired and analyzed. I put some questions to Alexander and then went away and thought about it.

I asked myself a seemingly simple question. What was the probability of the Spicers and this misidentified otter coming together in such a way to produce this effect? Or to put it more precisely - Nessie hit the news in April 1933 and the Spicers had their experience four months later in July 1933. What was the probability of otter, car and conditions confluencing in this miragical manner within the 81 days between the first public announcement of a sighting (the May 2nd edition of the Inverness Courier and the Spicers' encounter on July 22nd)?

I did some sums and came up with my own answers. The odds against it happening in that four months was about 2330 to 1 against. In other words, it was not at all likely to have happened. I have laid out the calculations at the end of this post for those who may want to suggest improvements in my assumptions.

I would add that changing the figures to lower the probabilities will present another problem to the skeptic. If one make the event more probable then over 77 years why have there not been more such misidentifications on land?

Now the retort may be that this is still more likely than a lumbering antediluvian crossing a road. But isn't that what the debate is all about? If you believe there is a large entity in Loch Ness then a land excursion moves from the impossible to the probable (depending on what you think it is). If you don't believe in such an entity, obviously no road crossing will ever occur.

But let us learn this lesson. Something may be plausible, but then ask yourself - is it probable?


Calculate the probability of seeing an otter at the right position on the road
at the right distance in the right weather conditions between April and July 1933!

1. Otter Population around Loch Ness

Let us assume an even distribution of otters around Scotland's shores and a recent article estimated 7,000 otters live in Scotland.

Coastline of Scotland = 11800km - 16491km (depending how accurate you wish to be)
Coastline of Loch Ness = say 1% of Scottish coastline
7000 otters in Scotland - say 1% proportionately at Loch Ness = 70

The otter population suffered under hunting and pollution but has been on the increase in recent years so double for larger otter population in 1930s = 140

2. Number of times an otter crosses the road on Loch Ness during day = 0.5

This is not so easy to assume. Most otter activity is at dawn/dusk and near water courses so a 4pm crossing near dry ground is most unlikely. But for the sake of this study we will assume an even distribution of otter activity but a daytime crossing is still a rare event.

3. Road perimeter of Loch Ness = 85km = 85000m

Eliminate 55% off road which does not pass Loch side = 38250m
35% is Between Fort Augustus and Foyers 10% is around Drumnadrochit
Eliminate 10% which is too high above shore for otters to traverse
A lot of otters are never seen because the roads are not near the shore or the road is high above the water.

So drop the number of otters to account for the lack of road:
Correspondingly drop number of otters = 77

4. How much of the remaining road is conducive to the special mirage conditions of the article?

The requirement is a gently ascending road over 100-200m. I have driven around the Loch quite a few times and a lot of road is bends, long stretches. So not a lot of opportunity but we will be generous and say:
Number of "mirage points" on perimeter (i.e. a hump on an undulating road) = 1
per 4km = 10 overall

5. Where the otter lies is crucial.

If it is too far from the dip horizon on either side then the mirage will not happen. That is why the range is only 1m or +/- 0.5m either side:
Otter has to be laterally within 1 metres of the "mirage zone" to work or 10 * 1m
zones in total = 10m

6. The otter has to be at beginning of "run" to have the desired effect

Or the sighting will be over in a flash. It is no use being half way or near the end at point of first sight:
Probability of otter about to cross the road = length of otter / length of road = 1.1/3.3 = 0.33

So combining all these "otter" factors together:
Probability that one of the 77 otters will cross one such area in one daylight
period = 77 * 0.5 * 0.33 * 10/38250 = 0.0033

7. What proportion of the daylight hours will be best for mirage conditions?

If it is nearing dawn/dusk then the temperature difference between the road surface and air above will not be high enough so for an average 12 hours of daylight deduct 6 hours:

Probability of good daylight mirage hours = 6/12 = 0.5

So otter crossing mirage area probability reduces to 0.5 * 0.0033 =
0.0017 in one day.

Now onto the other party - the car driver.

8. Assume car has to be at a dip approaching this hump for mirage to be effective.

Assume number of dips = number of humps = 10

9. Monster in sight for a "few seconds"

So at 9m/s zone = 9 * 3 = 27m
Car observer has to be within 27m of the dip zone for mirage to be effective or a total zone of 27 * 10 = 270m

That is the practical maximum but what was the actual range for an effective mirage? The small angle for an effective mirage suggest not much so let's third it: 9 * 10m = 90m

10. To see the whole mirage for a few seconds suggests the car has to be just entering the mirage zone

So probability that observer is at start of "mirage zone" = 1m/9m = 0.11

Probability that one car will be in this zone at any time = 90/38250 * 0.11 =

11. Of course, it is the number of cars passing these optimal mirage points that counts so:

Number of cars on south side of road crossing these points in one day = one every ten minutes = 36 during the 6 hour mirage time window

This is based on my own observation in July 2010 between Dores and Whitefield
where a car passed every minute and the assumption that car ownership is 10 times more than what it was in 1933.

Probability that a car will be over one of these mirage dips over 6 hours at any

one time = 36 * 0.00027 = 0.0097

Probability that otter at hump zone and car at dip zone will coincide =

0.0097 * 0.0017 = 0.000016

12. Probability that weather will be hot, no clouds, no shade, no recent rain

= 1 in 3 = 0.33

Overall probability on one day = 0.0000053 or 189393/1 against

13. Sum up over entire period
in question

81 days = 81 * 0.0000053 = 0.00043 or 2329/1 against


  1. You need to multiply the inherent probability, which you've calculated, by the number of possible occasions on which it might have happened: frequency of driving along the road, for example, and over what period of time it would have aroused Spicer-like interest, certainly a period of years if not decades. In that case, odds of 1500 to 1 are not very long at all.
    Furthermore, an important principle in dealing with unlikely or anomalous claims: if something happened, then it CAN happen, no matter how improbable

  2. I'm curious, Glasgow Boy. Are you familiar with the game of golf? I would assume so. Could you do a similar calculation for the chances of a hole-in-one on a certain hole at a golf course? I'm sure you should come up with chances far less that 1/1500...

    And yet there are holes-in-one every day. Sometimes people even get two hole-in-ones on the same day! That's double impossible! What does it mean?

    You also don't impress me with your attempts at logic. How did you get from Lock Ness being "1% of the coastline of Scotland" to otter population? Eurasian otters live in fresh water, what does the sea coast of Scotland have to do with anything? And why would they be evenly distributed? Obviously more otters live in larger lakes, and Loch Ness being one of the larger lakes would probably have a larger otter population. You just pulled a number out of your ass, and you like it because it's small, but it has nothing to do with reality. I could tear apart every bullcrap number you made up from that point on as well, but let's face it, there's very little point. A house built on sand will not long stand.

  3. You have committed what is one of the many things referred to as the "prosecutor's fallacy". Seeing miraged otters is highly improbable but seeing monsters is improbable too. You need to compare the probability of the Spicers seeing a otter to the probability of them seeing a large unknown animal on land. I am not sure either probability can be calculated with accuracy but given the sparsity of land based Loch Ness sightings over the years I imagine the probability of seeing a Loch Ness monster on land is pretty low too.

    As an aside I cannot comment on the rest of your calculations but I don't understand the use of coastline at stage 1. Presumably the Scottish coastline figure refers to sea coastline not the side of freshwater lochs so I don't see how that can be used to calculate the otter pop of Loch Ness.

  4. Steuart Campbell4 October 2010 at 03:14

    Henry is correct and your calculation is ludicrous. Otters do (did) live in L Ness, so it is 'possible' for one to have crossed a road (they were quieter then than now). So it 'possible' for it to have been seen by the Spicers, even if that coincidence was rare (the odds and the 'probability' are irrelevant). However, as Aleks (sic) knows, I am sceptical about there being a mirage; I'm even sceptical about the Spicer's report.

  5. Henry, that number already caters for multiple cars over a 3 month period. The odds stated are for a 1930s flow of cars over a given number of optimal mirage zones over 3 months. The odds of one car at any given time was much, much less.

  6. Scott,

    The equivalent of "Holes in one" over a given number of "golfers" is already factored in.

    Otters can live on coastal areas no problem at all - they have been seen. The only requirement is access to the river/stream outlets when their foraging is over.

  7. I wonder if any properly read what I posted?! I don't blame anyone for not doing so - it is a bit drawn out.

    Steuart, it is not a matter of quietness for otters. They are dawn/dusk creatures by instinct and they stick close to water courses near roads.

    If we leave aside the improbability of a miraged otter (which you seem to agree with me on), I don't think the Spicers would be fooled by an otter. But I might be wrong. If otters were so easily misidentified, I would expect a lot more "land sightings".

  8. Anonymous,

    Well that's the nub of the argument, isn't it? Is there a large entity in the loch or not? The answer (or one's belief) colours how we view otter et al explanations.

    What are the odds of a large entity/entities in the loch? Perhaps someone versed in Loch Ness flora and fauna could answer that. However, it would require multiplt answers depending on what you believed Nessie to be.

  9. No,it should not colour the probs with regard to this exercise. Two distinct but precise probabilities need to be compared. I think I was a bit unclear stating the probabilities.
    They are
    1. What is the probability of the Spicers seeing a miraged otter. I am not convinced of all your calculations but I can see that this number is obtainable
    2. What is the probability of the Spicers seeing a monster (given there really is a monster in the lake). The parenthetical statement is not a statement of beliefs but part of the specification of the probability i.e. what is the probability of seeing the monster knowing that there really is a monster in the lake
    (whether or not it actually does exist is an irrelevance at this stage).

    The trouble is I cannot think of a way to estimate the latter probability.

    Your current argument is equivalent to saying the probability of two cot deaths in a family is very small therefore the mother of the two children who have died seemingly from cot deaths is a murderer whereas as the Royal Statistical Society pointed out we need to compare the probability of two cot deaths in a family against the probability of a mother being a double murderer. Check out the Sally Clarke case.

  10. I have revised the odds calculated based on a more realistic "monster expectation" period starting from the first newspaper account on May 2nd 1933 rather than when the actual date of the Mackay sighting which was essentially unknown to the world. So we go from 120 days to 81 days.