Independence and Combining probabilities.

As I alluded to in my previous post, simple probability is really a matter of observation, to produce a model. If you roll a six sided die over and over again, you’ll see that each face comes up pretty much equally often, and so you model it as 1/6 probability for each face. There’s not really a huge amount of principle behind the basic models: it’s really just whatever works. This is the root of the distinction between interpretations: a frequentist starts with an experiment, and builds a descriptive model based on it, and says that the underlying phenomena being tested has the model as g, a property; a Bayesian does almost the same thing, but says that the model describes the state of their knowledge.

Where probability starts to become interesting in when you combine things. I know the probability of outcomes for rolling one die: how can I use that to see what happens when I roll five dice together? I know the probability of drawing a specific card from a deck: what are the odds of being dealt a particular poker hand?

We’ll start with the easiest part: combining independent probabilities. The probability of two events are independent when there’s no way for the outcome of one to influence the outcome of the other. For example, if you’re flipping a coin several times, the result of one coin flip has no effect on the result of a subsequent flip. On the other hand, dealing 10 cards from a deck is a sequence of dependent events: once you’ve dealt one card, the next deal must come from the remaining cards: you can’t deal the same card twice.

If you know the probability space of your trials, then recognizing an independent situation is easy: if the outcome of one trial doesn’t alter the probability space of other trials, then they’re independent.

Look back at the coin flip example: we know what the probability space of a coin flip looks like: it’s got two, equally probable outcomes. If you’ve flipped a coin once, and you’re going to flip another coin, the result of the first flip can’t do anything that alters the probability space of a subsequent flip.

But if you think about dealing cards, that’s not true. With a standard deck of cards, the initial probability space has 52 outcomes, each of which is equally likely. So the odds of being dealt the 5 of spades is exactly 1/52.

Now, suppose that you got lucky, and you did get dealt the 5 of spades on the first card. What’s the probability of being dealt the 5 of spades’s on the second? If they were independent events, it would still be 1/52. But once you’ve dealt one card, you can’t deal it again. The probability of being dealt the 5 of spades as the second card is 0: it’s impossible. The probability space only has 51 possible outcomes, and the 5 of spades is not one of them. The space has changed. That’s the definition of a dependent event.

When you’re faced with dependent probabilities, you need to figure out how the probability space will be changed, and incorporate that into your computation. Once you’ve incorporated the change in the probability space of the second test, then you’ve got a new independent probability, and you can combine them. Figuring out how to alter the probability space can be extremely difficult, but that’s what makes it interesting.

When you’re dealing with independent events, it’s easy to combine them. There are two basic ways of combining event probabilities,
and they should be familiar from logic: event1 AND event2, and event1 OR event2.

Suppose you’re looking at two test with independent outcomes. I know that the probability of event e is P(e), and the probability of event f is P(f) Then the outcome of e & f – that is, of having e as the outcome of the first trial, and f as the outcome of the second, is P(e)×P(f). The odds of rolling HTTH on a coin is (1/2)*(1/2)*(1/2)*(1/2)=(1/16).

If you’re looking at independent alternatives – that is, the probability of e OR F, you combine the probabilities of the event with addition: P(e) + P(f). So, the odds of drawing any heart from a deck: for each card, it’s 1/52. There are thirteen different hearts. So the odds of drawing a red are 1/52 + 1/52 + … = 13/52 = 1/4.

That still doesn’t get us to the really interesting stuff. We still can’t quite work out something like the odds of being dealt a flush. To get there, we need to learn some combinatorics, which will allow us to formulate the probability spaces that we need for an interesting probability.

Probability Spaces

Sorry for the slowness of the blog lately. I finally got myself back onto a semi-regular schedule when I posted about the Adria Richards affair, and that really blew up. The amount of vicious, hateful bile that showed up, both in comments (which I moderated) and in my email was truly astonishing. I’ve written things which pissed people off before, and I’ve gotten at least my fair share of hatemail. But nothing I’ve written before came close to preparing me for the kind of unbounded hatred that came in response to that post.

I really needed some time away from the blog after that.

Anyway, I’m back, and it’s time to get on with some discrete probability theory!

I’ve already written a bit about interpretations of probability. But I haven’t said anything about what probability means formally. When I say that the probability of rolling a 3 with a pair of fair six-sided dice is 1/18, how do I know that? Where did that 1/6th figure come from?

The answer lies in something called a probability space. I’m going to explain the probability space in frequentist terms, because I think that that’s easiest, but there is (of course) an equivalent Bayesian description.) Suppose I’m looking at a particular experiment. In classic mathematical form, a probability space consists of three components (Ω, E, P), where:

1. Ω, called the sample space, is a set containing all possible outcomes of the experiment. For a pair of dice, Ω would be the set of all possible rolls: {(1,1), (1,2), (1,3), (1,4), (1,5), (1, 6), (2,1), …, (6, 5), (6,6)}.
2. E is an equivalence relation over Ω, which partitions Ω into a set of events. Each event is a set of outcomes that are equivalent. For rolling a pair of dice, an event is a total – each event is the set of outcomes that have the same total. For the event “3” (meaning a roll that totalled three), the set would be {(1, 2), (2, 1)}.
3. P is a probability assignment. For each event e in E, P(e) is a value between 0 and 1, where:

$Sigma_{ein E} P(e) = 1$

(That is, the sum of the probabilities of all of the possible events in the space is exactly 1.)

The probability of an event e being the outcome of a trial is P(e).

So the probability of any particular event as the result of a trial is a number between 0 and 1. What’s it mean? If the probability of event e is p, then if we repeat the trial N times, we expect N*p of those trials to have e as their result. If the probability of e is 1/4, and we repeat the trial 100 times, we’d expect e to be the result 25 times.

But in an important sense, that’s a cop-out. We’ve defined probability in terms of this abstract model, where the third component is the probability. Isn’t that circular?

Not really. For a given trial, we create the probability assignment by observation and/or analysis. The important point is that this is really just a bare minimum starting point. What we really care about in probability isn’t the change associated with a single, simple, atomic event. What we want to do is take the probability associated with a group of single events, and use our understanding of that to allow us to explore a complex event.

If I give you a well-shuffled deck of cards, it’s easy to show that the odds of drawing the 3 of diamonds is 1/52. What we want to do with probability is things like ask: What are the odds of being dealt a flush in a poker hand?

The construction of a probability space gives us a well-defined platform to use for building probabilistic models of more interesting things. Give a probability space of two single dice, we can combine them together to create the probability space of the two dice rolled together. Given the probability space of a pair of dice, we can construct the probability space of a game of craps. And so on.