Probability for Wargames 13 - more Chain of Command

Derek raised an interesting point re last time's article:

"For example you want to work out the "chance of activating a team with a leader present".

The team itself can be activated if you roll one or more 1s with 5D6.

You can activate a Junior Leader by rolling any combination of dice that add up to three. That's one or more 3s, three rolls of 1, or one or more 2s, plus one or more 1s.

If there's a Senior Leader hanging about you can also activate the team with any combination of dice that add up to four.

It's all horribly complicated."

Yes it is :D However, in the case I was raising, that of activating a team, adding a 1 and 2, or 3 1s to activate the Junior Leader makes no difference to getting the team activated, since you can't do that without rolling a 1. But it does improve the odds on (say) being able to rally off a point of shock first :D

However, as our final piece in this series, at least for now, let's look at the really complicated one:

What are the odds on not activating a section (i.e. on a 2) in the presence of its Junior Leader (on 3) and a Senior Leader (on a 4)?

A 2, 3, or 4 will do: the odds of 5 rolls. none of which contain a  2, 3, or 4 are 1/2 ^ 5, or one in 32, or roughly 3.1% (by now you should be able to figure out how we get there), so the odds of at least one 2, 3, or 4 are 96.9%. In fact, we can skip this bit, but let's note that number anyway.

The only other roll that will get us the unit activated is to roll two or more 1s that add up to 2 or more. (Any other result that adds up to 2, 3, or 4 will already contain a 2, 3 or 4, and have been counted in the above 96.9%).

So, we need the odds on rolling at least two 1s and no 2s,3s or 4s. Odds on rolling a 1 are 1/6, on a 5 or a 6 are 2 in 6 or 1/3.

Odds of rolling all 5's,6's = 1/3 ^ 5, which is 0.41%
Odds of rolling one 1 and the rest 5s & 6s = 1/3 * 1/3 * 1/3 * 1/3 * 1/6 * ₅C₁, or about 1%
Odds of rolling 2 1s ... = also 1% (I'm not showing my working :D)
Odds of rolling 3 1s ... = roughly 0.5%
Odds of rolling 4 1s ... = roughly 0.13%
Odds of rolling 5 1s ... = 0.01%

A quick sanity check - all those should add up to our outstanding 3.1% which they do, as near as damnit (which is why that earlier calculation was useful).

Of course, by the time you've got to here, like me you should have realised that the only roll that will fail to activate our unit is to roll all 5s and 6s and no more than one 1. Odds, around 1.41%.

If you're bored, try a similar calculation for the same without the Senior Leader :D

What are the odds on a extra phase if you have only 4 command dice? How about 6?

If you've followed this series this far, you should be able to work it out. [Hint, you'll need to work out the cumulative odds of rolling k sixes out of n, using _nC_k.] I'll leave it as an exercise for the eager reader: your prize will be a namecheck in the wrapup post for this series.