In Uncertainty Wednesday today we continue our analysis of the Zoltar example, which I introduced two weeks ago. We came up with the super simple expression B + G = N, where B is the number of Bad fortunes, G is the number of Good fortunes and N is the number of all fortunes. Now while this may still strike you as simply an accounting identity, let’s consider the case of N = 1. Because these have to be integers, it must be the case that either (B = 1 and G = 0) or (B = 0 and G = 1). Why do they have to be integers? Because that is what we observe. We observe a complete fortune being printed and it is either a Bad or a Good one.
So when the machine starts, i.e. when N = 0, what should we expect for the first fortune? It must be either B or G and we see now how our explanation mathematically captures this split. But how should we think about this “either B or G” in advance? This is where we first encounter the notion of probability in this series (I know, it has been a long time coming, sorry!). We could assume that getting a B or a G are equally likely and express that as
P(B) = 0.5 and P(G) = 0.5
where P(B) stand for the probability of observing B and we use a number between 0 and 1 (0.5 is much easier to work with mathematically in a formula instead of 50% – the % sign really just stands for /100, i.e. divide by 100).
Note that P(B) + P(G) = 1, i.e. the two probabilities added together equal 1 (or 100%). It has to be that way because our explanation says that B and G are the only two possibilities. So if we think there is a 50% chance of observing B, then there has to be a 50% chance of observing G. So in that case why would we pick 50:50 odds? Why not say P(B) = 0.2 and P(G) = 0.8? After all, that too adds up to 1?
To understand that let’s focus on what we mean by “the machine starts.” We mean that we have no *prior* observations. This could be either because we do not have any or because the machine has a *reset* button that we assume does start the machine anew (as described before this is in itself a source of uncertainty that we have chosen to ignore in our current explanation).
But how does having no prior observations translate into assuming 50:50? Because this is now all we have to work with. Our explanation is simply B + G = N and we have no observations. Now it could be different. For instance, someone could walk up to us and say “I was here yesterday and roughly 1 out 5 people got a Bad fortune” (this means we have prior observations). Or we could be in a situation where there is a label on the machine that says “Prints 20% Bad fortunes.” We could think of the latter as an additional piece of explanation, which we could capture as P(B) = 0.2.
In the absence though of either prior observations or a better explanation, the 50:50 even split between the two captures the idea that we have *no* prior beliefs about which fortune is more likely to be observed. A different way of saying that is that the even split expresses the highest degree of uncertainty about what we will observe. We will eventually make this more mathematically precise, but for now this is all about developing an intuition for these statements. So why is it that 50:50 captures more uncertainty than say 20:80? Imagine betting with me about whether the first fortune will be B or G? If it is B I pay you and if it is G you pay me. What would you consider to be a fair bet? Well, if your prior belief is 50:50 then it would be fair if we each pay the same amount. But if it is 20:80 then you would never accept that because you would figure that in 20% of cases (B) I pay you and in 80% of cases (G) you pay me. So you would never accept it being the same amount in either case. This shows that you now have some degree of knowledge (or reduced uncertainty).
We will continue next week with looking at what happens after we have made the first observation.
