Numbers, numbers and even more numbers...

Bang on gunner! I reckon 47.5% chance of a hit after mods for stabilisation!

I have been enjoying posts by top boffin Phil Dutre on his blog Wargame Mechanics. Not being a prof, and having missed negative binomial distribution in school (I forgot my PE kit, honest), I have to take the posts very slowly. Nevertheless, there is some great stuff here.

One of the issues with many relatively simple mechanics is figuring out the parameters, what sort of results spread do I really want? This is the case for any rules where there is a roll to hit and the target is destroyed after an appropriate number of hits. Phil examines this issue in detail in his July post "1 hit for 10 damage, or 10 hits for 1 damage each?".

I strongly recommend you read his post, which has graphs and lots of squiggly equations. However, I have simplified the core of his presentation down to this:

1. What is the hit probability? You are shooting at a target and you hit on a score of 4, 5 or 6 on one d6. This gives 3 possible successes from 6 possible outcomes so 50% probability. 

Note that if this was a d8 and the same hit number is used (4 or more) the probability of success is 62.5% (5 successful possible outcomes out of 8 possible outcomes). There is great potential for using different types of dice.

2. How many hits can the target can take before destruction? Let's say 4 points of damage.

3. How much damage is caused by one hit? Keep it simple and say one point of damage per hit.

So how do we calculate the number of shots that are required to kill the target?

Shots to kill = target damage points/(hit probability x damage per hit)

In this scenario the shots to kill = 4/(3/6 x 1)

Therefore shots to kill = 4/0.5 = 8

This is easy peasie and you can set up an excel spreadsheet to work out the range of results with different parameters. 

The more tricky issues to consider are the likely number of units shooting at the same target in one turn, the number of units and the number of turns in the game. This allows you to consider what sort of attrition rates you need to have a decent and exciting game in a useful number of turns.

Lessons here are don't forget your kit when doing posh maths, don't drink beer and do hard sums and sometimes doing some proper maths rather than endless play testing will help with design decisions. I have, of course, learnt none of these but am continuing to aim to be a better person!

Edit: Many thanks to everyone for helping me with my maths homework. Hopefully this is now correct!

Normandy numbers (1)

Pass the ammunition

One of the subjects I continually return to is Normandy. Not just D-Day but the campaign through to the liberation of Paris. It seems to me that the battle for the history of the campaign is as bitter as the actual fighting. If we leave aside the French and Germans (which we really shouldn't), we have an on going three way battle between the Americans, British and Canadians about who did what, how good they were (or not) and how bad Monty was.

My motto is "when the going gets tough the tough get the numbers and do the hard yards". There is a lot of good data about Normandy and our understanding of the campaign will be enhanced (even if the arguments are not resolved) by its full and sensible use.

I have recently been re-reading Max Hastings' book "Overlord" (Pan 1984). Appendix 5 contains "Some British Administrative Statistics" which I assume are from 21 Army Group. The data is familiar to me but I can't pin down the precise source. Probably not the best place to start a crusade for fact based history but certainly it is a first step. If anyone knows where this data originally came from, please let me know.

The data is for the period 12 June 1944 onwards. It gives rounds per month by gun type and rounds per gun per day. I have analysed this data and drawn up some graphs.

This first graph shows the rounds used per month. June numbers are smaller because the data is from 12th only. However, what started my interest is the large amount of mortar ammunition used in June. I assume because the artillery build-up, over the beaches, took time.

To take a closer look at the numbers I analysed the percentage contributions from each gun type. Mortar rounds are nearly 30% of total ammo use in June. 

I then developed some "effective fire" values using some very helpful data I found at:

http://nigelef.tripod.com/wt_of_fire.htm.

 

The effective fire values are based on the square root of the weight (in kgs) of the HE content of the shells. This shows, for a given weapon type, the relative effectiveness of its fire. Example values are 0.7 for a 3" mortar, 0.9 for a 25-pdr, 2.3 for a 5.5" gun and 3.6 for a 7.2" gun.

 

What I find really interesting about this data is the weight of fire contributed throughout the campaign by a fairly small number of medium and heavy guns (~450 guns out of some 2,500 total including mortars) but also the contribution made by divisional 3" and 4.2" mortars.

Time to find some data on American ammunition usage.....