## Monday, 1 October 2012

### Probability for Wargamers 4 - More on looting

So - last time's homework problem. If you succeed on a 6, and fail completely on a 1, how many rolls do you need to make before your chance of success exceeds 50%?

The answer, which may or may not surprise you - it did me till I thought about it - is an infinite number! And you actually don't need to do much maths to prove it.

Think about it: the probability of failing or succeeding, on any given roll, is exactly the same, 1/6. You are as likely to find nothing at all (and give up) as you are to find something, and have odds of 4 in 6 of rolling again. So the odds of rolling again after two throws is 4/6 * 4/6, after three is 4/6 * 4/6 * 4/6: this number gets progressively smaller and smaller really quite fast (it's less than 1 in 10 after 6 rolls) as you roll more dice, but never quite reaches 0%. Which means your chance of stopping never quite reaches 100%, and if you do stop, it's a straight two-way bet whether you stop because you succeed, or because fail completely, so your chance of success never quite reaches 50%. Full marks if you figured that out!

The rules also say that if you don't find anything in the first two buildings, you will find something in the third, and that you need to find two somethings to succeed. I wonder - if you have infinite time to spend, what odds do you have on successfully completing the raid?

Again that's pretty easy to figure out. You have a 1/2 chance of finding something in each of the first two (as you'll eventually roll either a 6 or a 1), and a 100% in the third if you need it, as you will eventually roll a 6. So in fact, the problem boils down to finding at least one in the first two buildings. What are the odds on that?

If things clicked for you in the past couple of posts, you'll realise it's easy - one minus the odds of finding none. Which is to say, 1 - (1/2 * 1/2), or 75%.

There are two things to add to this. First off, it should be pretty obvious from this that your best strategy is to search in parallel... like I didn't... with as many groups as you can spare, and defend your searchers with the remainder of your force. Remember the plan.

Secondly... remember the gambler's fallacy! Dice don't have memory. Just because we figured out that your chance of rolling a 6 in four dice rolls is better than 50%, that doesn't mean that if you blow the first roll, you have a just over 50% chance of getting it in the next three! The first roll is history: it's a statistically independent event whose odds of failure used to be 5 out of 6 until you rolled the dice and failed, but are now 100%. Your odds for any single roll are still 1 in 6, no matter what you rolled last time.

Next time, we'll roll 2d6, and look at Black Powder and Blitzkrieg Commander command rolls.