Let's go back a couple of weeks to the last Dux Britanniarum battle Andy and I fought. Chris Stoesen commented on the battle report that it was a shame I didn't get the loot, and I was prompted to wonder, just how hard is it to get the loot?
Basically, to get the loot, you need to roll a 6, so how many rolls is it typically going to take? Let's rephrase that as 'how many times do you have to roll before you have a better than even chance of having succeeded?'
Obviously, on the first roll, you have a 1/6 chance of success, and therefore a 5/6, or 83.33%, chance of failure. What's our chance of success on exactly the second roll?
Actually, that's pretty easy. We need to fail on the first roll, and succeed on the second. That tell-tale "and" hints that we should multiply, so that's 5/6 * 1/6, or 5/36, which works out to 13.89%. Our chance of succeeding in no more than two rolls, then, is the chance of succeeding on the first roll, PLUS the chance of succeeding on the second having failed on the first, as these are mutually exclusive events (yes they are - think about it). That's 1/6 + 5/36, or 11/36, which is 30.56%.
But actually, there's an easier way, because we know the odds of failing twice - that's the odds of failing on the first roll AND the second. 5/6 * 5/6, or 25/36, or 69.44%. So the odds of not failing twice (i.e. succeeding on one of the first two rolls) is one minus that - 30.56%, as before. By extension, the odds of succeeding in three rolls or less is one minus the odds of failing all three, or (1 - 5/6*5/6*5/6).
I have to admit, at this point I bunged it in a spreadsheet, and it turns out, after filling down a few rows, that for a chance of succeeding more than half the time, you only need to make four rolls - 51.77%, to be exact.
Right - homework for this post. The rules actually say that if you roll a 1, there is no loot to find, and you can stop rolling dice. How does that change the odds? How many dice do you need to roll for a better than 50% chance of success now? The best method of attack might be to work out the chance of getting to roll again next time, and proceeding from there.
The answer will be in the next post, and may surprise you. I suspect it may surprise Mr. Clarke, too :D