As those of you who are familiar with the mechanic will know, the Too Fat Lardies games rely on a card-based activation mechanism, and in most of those, the activation sequence is ended when a Tea Break (or whatever it's named for that particular ruleset) card is drawn.
So. What's the likely percentage of cards that come up before the Tea Break card comes up?
This seems quite tricky at first glance, but here's one way. The Tea Break card could be anywhere with equal probability. If it's first of N cards, then no useful cards get drawn before it, and if it's last, then all of the (N-1) useful cards get drawn. Simple intuition says the answer, the average probability, is therefore 50%.
More rigorously (and so we can wave similar results around with aplomb if we need to later!) - if the Tea Break card is drawn at position M, there are M-1 cards before it, which is (M-1)/(N-1) of the whole deck of useful cards. So the average is the sum of all those (M-1)/(N-1), with M ranging from 1 to N, divided by the N possible positions.
(And here I wish I had a maths font.)
The sum of all the numbers from 0 to N is, by a handy mathematical co-incidence, N*(N+1)/2 (check it if you don't believe me). So the sum of all the individual (M-1)/(N-1) for M from 1 to N (i.e. M-1 from 0 to N-1) is ((N-1)*N/2)/(N-1), which boils down to N/2. And if you divide that by N to get the average, you get 1/2, or 50%. Yay.So, the probability of any random card appearing before the Tea Break card is 0.5, or 50%. Let's look at what this means for the question of blinds.
At the start of the game, there are three cards in the deck: Allied Blinds, Axis Blinds and the inevitable Tea Break. And there's a 50% chance that your Allied blinds card will come up before the Tea Break.
Let's activate an Allied platoon and its Big man, so now our deck contains Allied Big Man 1, Allied Platoon 1, Allied Blinds, Axis Blinds and Tea Break. Chances of the Allied Blinds coming up and you getting to move the rest of your blinds? Still 50%. But what are the odds on Allied Platoon 1 getting to activate?
Well - in order for no part of the platoon to activate, neither the Platoon card nor the Big Man card must come up. The probability of either of those events not happening is 1/2, so the probability of both cards not coming up is 1/2 * 1/2, or 1/4, 25%. So the chance of some part of Allied Platoon 1 getting activated is 3/4 - 75%. If there were no Big Man card for it, it'd be 1/2, 50%, the same as for the blinds.
In fact, given most units in IABSM tend to come with an associated Big Man, once you've deployed a unit off blinds, its chance of getting activated either by itself or its Big Man, is 75%, compared to only 50% if it stays on blinds. Admittedly, unless it's a level 3 Big Man, all its sections won't necessarily activate if it's a platoon, but...
In other words, the chance of units remaining on blinds getting activated doesn't decrease, but it is statistically better to deploy if you want to improve the odds of at least one section of a platoon (say) activating.
This article's homework? Prove, with a reasonable degree of rigour, that adding a second Tea Break card means you draw on average 2/3 of the deck before the second Tea Break comes up.