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