Gandalf's thread of numbers and probability

[Gandalf]

People have said I talk too much about numbers and probability. But I do like talking about them,. I've always liked numbers since I was a kid, and I have a very mathematical brain (partly why I'm a computer programmer). Anyway, a solution to this is for me to post more about numbers in this thread, and less in other threads. That way I can waffle about numbers and you can all ignore this thread if you want

Alternatively, if you are interested, post problems here & I will work cheerfully do the sums

[Gandalf]

First up - the Bloodthirster! He's the big guy of a Khorne team and has all manner of skills. I faced him yesterday and he was almost always used to blitz. But how effective is he?

Problem - what are the chances of a blitzing bloodthirster KO'ing or injuring a player on any given turn?
Assumptions
- Any failures are not re-rolled (doing so is risky as he has Loner)
- All blocks are 2 dice ones (in yesterday's game, 16 of 18 were 2D blocks)
- Neither player involved has any block-related skills, apart from the bloodthirster's in-built ones
- Opponent is AV7 or higher. Due to claw it doesn't matter what the AV is as long it's 7 or more.
Notes
- The sums are "per turn" rather than "per block" as he has frenzy.

Working-out
Chance of knocking over a player. Wild Animal needs to be passed (2+) and a defender stumbles/defender down needs to be rolled in either the first or second block. Due to juggernaught, only a double both-down will end the turn.
This works out to be 65.59%

Given a successful block, what is the chance of a KO or injury? Mighty blow & Claw means a 58.3% chance of breaking armour. The maths for the injury roll are slightly more complex as it depends if Mighty Blow was needed to break the armour or not, but the final sums show a 16.9% chance of a KO or a 14.35% chance of an injury.

The answer
Given the above assumptions, If a Bloodhtister blitzes, there is a 65.59% chance of the target player ending up on the floor.
This breaks down into a 9.41% chance of an injury..
.... or 11.08% chance of a KO.
17.76% stunned, 27.33% just knocked over.
Or to put it another way, about 1 in 5 blitzes will remove the target from the pitch.

[notjarvis]

What Armour value are you assuming for the defender?

nm. He's got claw Ain't he

[Gandalf]

What Armour value are you assuming for the defender?

- Opponent is AV7 or higher. Due to claw it doesn't matter what the AV is as long it's 7 or more.

[Red Lion Three]

Dear Gandalf,

here is a practical Blood Bowl statistics project for you which could benefit the whole community:

Quite often a game is interrupted by a simple disconnect or the red text of doom, and sometimes it is too late to have another go at the game. I'm thinking this is a bit like rain during a one day game. Could you work on a Duckworth-Lewis system equivalent - lets call it the Dragonvalue Morse Protocol temporarily so its got a name - so one only has to play the remaining turns when one restarts, rather than a whole game?

It should keep you busy for hooouuurrsss...

RL3

[Raveen]

Bloody hell... that's quite the ask.

[Red Lion Three]

And not entirely serious...

[Raveen]

Be careful what you wish for...

[Gandalf]

There speaks two people who have known me for quite a while

[id3nt1ty]

Haha, cheers for the statistics Gandalf!

The one I struggled with was working out the probability of even getting to block a Nurgle Warrior with my Bloodthirster, as I'd need two 5/6 rolls to do that. In particular, I struggled to comprehend how that would result in the Bloodthirster failing to block 1/3 attempts. I was convinced la and Daigaro were wrong, but in the end I mathed myself into a corner where I had to accept the truth.

Probably the thing that confounded me the most was that I was trying to do 1/6 * 1/6, which appears to be 1/36 (the true probability of rolling two ones on a pair of dice at the same time (right?)). After going down a tangent of dependent vs. independent probability (dice rolls being independent), I realised I needed to look at it the other way around. 5/6 * 5/6 = 25/36 is very different to 1/36.

[Gandalf]

I love the phrase "mathed myself into a corner" Yes you have to be careful working out things like that, do you a) work out the chance of something happening or b) do 1 minus the chance of it not happening.... easy to get them confused. Maybe it helps to link your numbers back to their in-game meaning - by doing 1/6 * 1/6 you were working out the chance of failing a wild animal roll AND a foul appearance roll (which can't even happen anyway as if you fail the first you never get on to the second).

In the absence of any proper suggestions (sorry Red Lion 3), the next topic will be one of the favourite activities of teams last season - injuring snotlings!

[Raveen]

When working out multiple dice rolls like that I tend to draw out a tree with each branch's probability so I can track it back.

Or just use a bb odds calculator, there are plenty on Android.

[Gandalf]

Rav - I haven't seen a calculator that can deal with the sort of complicated multi-dice multi-skill situations I will try to be dealing with here, but if there is one do point it out.

So then, snotlings! The weakest and cheapest player in the game, at 20k, and a strength of 1. At least they have dodge to aid in their futile fight for survival.

Problem - what are the chances of getting a snotling off the pitch?
Assumptions
- All blocks are 3D blocks (a regular ST3 lineman vs. a snotling will give 3D block)
- The snotling does not have block or wrestle (requires doubles and living long enough to get it)
- Attacker does not roll triple red skull (1 in 196 chance) or nothing better than both down if he doesn't have block (1 in 27 chance). I suspect a team re-roll would be used in either of these cases anyway.
- If the attacker fails to knock over the snotling, it is not re-rolled (it's rarely worth it, unless he has the ball)


Scenarios to consider
- Attacker having no block-related skills
- Attacker having block or tackle (they are equal in their ability to get a snotling on the floor)
- Attacker having block AND tackle
- Attacker has mighty blow

Working out
Chance of knocking over a player. With no skills, there is only one side of the dice that will result in the snotling being knocked over. With 3 dice this is a 42.1% chance. This increases rapidly though if you have block or tackle, to 70.4% (1 minus 2 thirds cubed), or a massive 87.5% if you have both.

Chance of injuring a player - With a pathetic AV of 5, you only need a 6+ to break armour. This is a 26/36 chance, or 72.2%, or 83.3% if you have mighty blow! Having crunched the numbers, the overall chance of injuring one - given a successful block - is 12.04% without mighty blow, or 21.91% if you have mighty blow. KO chances are 18.06% or 24.8% chance respectively.

Overall ...
The chances of stunning/injuring a snotling, given the assumptions listed, are....
No skills..... 7.61% / 5.07%
Mighty blow... 10.47% / 9.23%
Block or tackle 12.71% / 8.47%
Block and tackle... 15.80% / 10.53%
Block or tackle, and mighty blow 17.48% / 15.42%
block, tackle and mighty blow 18.63% / 16.44%

Conclusions

Obviously, snotlings are going go get splattered, a lot. So have plenty in reserve.
When I lose 2 or 3 snotlings to unskilled players in 1 turn, I am justified in feeling unlucky!
Having block or tackle makes a huge difference, taking the depitch chance from 12.7% to 23.2%.
Mighty blow is traditionally seen as the injury-causing skill - and it does beat block/tackle in this case but only just. Block/tackle gives a better deptich chance than MB. However one of those skills plus MB beats both of those skills. It might be interesting to run similar numbers on a wider variety of cases - higher armour, less block dice etc.

[Raveen]

Using a calculator:

To injure:
3 die block followed by 6+ (armour) followed by 9+ (Injury with Stunty modifier) = 8.5%
3 die block with 1 skill followed by 6+ followed by 9+ = 14.1%
3 die block with both skills followed by 6+ followed by 9+ = 17.6%

To depitch:
3 die block followed by 6+ (armour) followed by 7+ (KO or better with Stunty modifier) = 17.7%
3 die block with 1 skill followed by 6+ followed by 7+ = 29.6%
3 die block with both skills followed by 6+ followed by 7+ = 36.9%

As Gandalf says, you're unlikely to reroll a block to hurt a Snotling, you'd only reroll in the case of a turnover so I haven't factored that in.

MB makes life a lot trickier so I can't be bothered to work that out just now but essentially it looks like a Snot leaves the pitch one time in every 3-6 blocks.

[Gandalf]

Hmmm I never knew that stunty made them easier to ko/injure. Hurrah for learning stuff. I'll have to redo the numbers to reflect that, once I get back to work & the spreadsheet I have.