This is going to be an occasional series on probability and statistics, and is largely going to set out to answer variants of the question 'just HOW hard was that?' or 'how ridiculously unlucky have I just been?'
For the first article, we'll go back to any number of games of WAB where I've demonstrated a prodigious ability to miss with the fair Cartimandua, Queen of the Brigantes. This one's quite useful, because it covers both of the two basic concepts of probability, namely when to add probabilities and when to multiply them.
So, to hit in WAB you need to roll equal to or greater than a certain number on a d6. If we want to get three or more, say, then four possible numbers on the dice will get us that: 3, 4, 5 or 6. Each of these has a 1/6 chance of coming up assuming the dice aren't loaded, and since these are mutually exclusive events (we can't roll a 3 AND a 6 on the same roll of one dice, for example), we can add the probabilities, which gives us a 4/6 chance, or 66.67%, of rolling 4 or less.
What's the odds of failing?
That's easy. We know that success and failure are a) mutually exclusive events and b) the only possible outcomes, so the total probability for them must add up to 1, or 100%. Hence the odds on failing are 100% minus the odds of success, i.e. 2/6, 1/3, or 33.33%. (Yes, obviously it's the 1/6 chance of rolling a 1 plus the 1/6 chance of rolling a 2, but sometimes you'll find this trick much easier.)
So: what's the chance of my barbarian Queen, with three attacks on a 3+, not hitting at all? That means I have to fail with the first dice, AND the second, AND the third. That phrasing is a dead giveaway - these are independent events (whatever your superstitions about dice), and this means that in order to figure out the odds of them all happening, the rules say we can multiply the probabilities together. So, the odds of failing three times are 1/3 * 1/3 * 1/3, or 1/27, which works out at 3.70%.
Man, my dice rolling sucks!