Basic Probability

As mentioned in class, we will learned a bit of probability this term. The most basic formula is as follows:

$$P(A) = \frac{\text{No. of outcomes for event A}}{\text{No. of possible outcomes}}$$

As simple as this is, it can sometimes be difficult to really understand the meaning of this formula. The mathematical notation is a bit strange, and there are hidden senses of logical conditions. It's also hard to understand the scope. The best we can do is just practice with it for well-known items.

So, the analysis of probability for a coin to come up heads vs. tails would look like this:

$$\text{Possible outcomes} = \{\text{heads}, \text{tails} \}$$

Just by counting everything that is in our set of possible outcomes, we can figure out the bottom of the formula:

$$P(\text{Coin = Heads}) = \frac{\text{No. of outcomes for Coin = Heads}}{2}$$

Now, the coin coming up heads is merely one of the two outcomes:

$$P(\text{Coin = Heads}) = \frac{1}{2} = 50\%$$

But what does this mean practically? The statement technically means that a long process of flipping coins will tend towards a distribution that looks even. It will only very rarely truly be that exactly 50% of the events will turn up heads, it will that if we flipped the coin for long enough, in ideal conditions, we should see a distribution that approaches 50% for each outcome.

It's really important to understand that meaning behind the formula, particularly for robotics, because conditions are rarely ideal, and we don't have forever to test a probability. Instead, we say that the probability is a model of what we might expect to happen, but expectation doesn't mean "we're really, really confident", it means, we have this process in mind, and a betting person should be careful.

So, with probability, we're always asking the question: if conditions were ideal, how could we set up a game of counting the outcomes we care about over all the outcomes that are possible? The posing of the question is hard to translate into math. For example, if I said, what's the probability that a signal is True a priori, you might say:

$$\text{Possible outcomes} = \{\text{True}, \text{False} \}$$ $$\text{Possible outcomes} = \{\text{True}, \text{False} \}$$ $$P(\text{Signal}=\text{True}) = \frac{1}{2}$$

But post-hoc, you might be dealing with a real experimental dataset that looks like:

Timestamp	Signal
00:00	True
00:01	True
00:02	True
00:03	True
00:04	False

In that case, just by counting up the events as they occurred, the true experimental probability was:

$$P(\text{Signal}=\text{True}) = \frac{4}{5}$$

So, our model did not agree with our experiment. Was it a bad model? No! But it just didn't account for some experimental reality. In this class, I will usually try to differentiate when we're talking about the analytical vs. experimental probability, often asking for both.