[r-t] Proportion of Surprise Methods

[Leigh Simpson]

> What proportion of TD minor grids are surprise methods?  And what
> proportion of regular or standard methods?  I think "a quarter" is too
> simplistic...

Well, a probabilistic argument could work as follows:

-----

When the treble hunts 2-3, the following place-notations are possible:

14
1456
1458
1478
145678
16
1678
18

When the treble hunts 4-5, the following place-notations are possible:

16
1778
18
1236
123678
1238
36
3678
38

When the treble hunts 6-7, we have the same situation as in 2-3 but
reflected about 4.5ths-place.

According to the CC decisions:

 > Surprise methods are Treble Dodging methods in which at least one
 > internal place is made at every cross section.

Considering any random, symmetric grid, and assuming that the changes at the
cross-sections are uniformly distributed and independent:

To get a treble-bob method, each cross-section must have the notation 18.
The probability of this is 1/8 * 1/9 * 1/8, i.e. 1/576 or 1.74e-3

To get a surprise method, each cross-section must have an internal place.
The probability of this is 7/8 * 8/9 * 7/8, i.e. 392/576 or 0.681

If we allow grids to be asymmetric, the probabilities change.

Treble-bob: 1/8 * 1/9 * 1/8 * 1/8 * 1/9 * 1/8, i.e. 1/331776 or 3.01e-6
Surprise: 7/8 * 8/9 * 7/8 * 7/8 * 8/9 * 7/8, i.e. 153664/331776 or 0.463

Further restrictions on what changes are allowed will alter the proportions
of method types. If we don't allow 7ths-place to be made above the treble,
the list of permitted notations is culled:

2-3:
----
14
1456
1458
16
18

4-5:
----
16
18
1236
1238
36
38

So we get (symmetric):

Treble-bob: 1/5 * 1/6 * 1/8, i.e. 1/240 or 4.17e-3
Surprise: 4/5 * 5/6 * 7/8, i.e. 140/240 or 0.583

-----

Obviously there's a couple of major flaws in this:

 1. This analysis in no way reflects what people might want to ring. Even
though treble-bob methods are constrained in terms of their grid structure,
people have rung quite a few of these. Any additional restrictions (e.g.
only triple-changes, plain-bob lead heads) will cause the probabilities to
be altered.
 2. It also assumes that the changes at the cross-sections are independent.
I'm not convinced that this is the case. Requiring the lead-heads to be
plain-bob, or even that there are seven leads in the course will almost
certainly mean that the assumption is incorrect.

Nevertheless it might be a useful start. I would suggest that it is likely
that the fraction of grids that are surprise is nearer to 1/2 than 1/4.

-----

Some ringers (e.g., I think, DJPJ) prefer the following definition of
surprise methods:

 > Surprise methods are Treble Dodging methods in which places are made
 > adjacent to the treble's path at every cross section.

This definition equivalent for minor methods, but diverges on higher stages.

This means that the proportions of valid changes alters:

2-3:
----
14 *
1456 *
1458 *
1478 *
145678 *
16
1678
18

4-5:
----
16
1778
18
1236 *
123678 *
1238
36 *
3678 *
38

Where the valid "surprise changes" are marked '*'. So the probability of
being a surprise method if one's grid is symmetric is now:

5/8 * 4/9 * 5/8, i.e. 100/576 or 0.174

-----

Leigh
(who is now waiting for RAS to say this is all a load of rubbish)