Posted by on Oct 22, 2007 in Richard and Judy - and probability | 11 comments

# Richard and Judy – and probability

Not withstanding Mike’s suggestion that I drop this topic as its not up to usual standard, I’m going to stick with it for a bit.

Jeremy says:

…the R&J show was basically scamming a significant proportion of the population into believing that they had a chance of winning… when they had NO chance at all at the time they decided to commit their money. It is in the withholding of this information that I think the unfairness lies.

I’m not sure that’s it, as, in my example where stage 1 of whittling down is that half the time entries are received they go straight in the bin, a significant proportion of the population have no chance of winning at the time they decide to commit their money (because their entry is immediately binned). Yet there is no unfairness in withholding this info from them.

I didn’t make my point about determinism clear, as some of you think it irrelevant. Well, may be it is irrelevant, but let me try again (as the point I was trying to illustrate is the crux of my question)!

I was trying to illustrate the point that what we consider “random” and “non-random” depends on how we set the parameters. Given the parameters include the laws of nature and all antecedent physical conditions, that this dice should now roll 6 is not random. There’s “no chance” of it doing anything else. On the other hand, given no more info than that the unloaded dice is given a vigorous unseen shake before being rolled in the normal way, it’s getting a 6 is random (there’s “just as good a chance it’ll roll a 5 or 4 etc.”).

We say the Richard and Judy case was unfair, and we say this because certain contestants – those who phoned after a certain point – were not excluded at random, but were ruled out from the start (so they had “no chance” of winning), whereas in my example where half the entries go in the bin, and the rest go into a random lottery, we say that those that went into the bin were fairly excluded at random: they did still have the “same chance of winning” as everyone else.

There’s a question about what we consider “random” and “non-random”, and thus fair (a “chance” of winning [even if very, very low, Mike!]) and unfair (“no chance” of winning at all), to which I don’t know the answer.

Why, in the RandJ competion, is it right to say certain entrants have “no chance of winning” whereas in my hypothetical competition those who phone in during times when entries go in the bin have “the same chance of winning as everyone else”?

[[N.B. Note this problem is not, Mike, that I cannot tell difference between very low, and no, probability. I do know about that (in methodology I teach the Bayesian response to the paradox of the ravens, which hinges on precisely that point).]]

1. A randomly selected entrant has the same chance of winning whether the winner is selected early or not.But this of course is a very low chance bearing in mind the cost of entry and the value of the prize. This seems to be the real unfairness.Perhaps, in the spirit of consumer protection, like on gaming slot machines, there should be some indication of the percentage payout for phone-in competitions.

3. I think this is an area that is very difficult to think about for most people and I certainly include myself.I wonder whether the problem at the core of this is that epistemic probability versus physical probability is one of those dualisms that hinders rather than helps.I think that under determinism, there is only epistemic probability because, from a physical perspective, there is no ‘likelihood’. But from a human point of view, even if we think determinism is false, it is not clear what indeterminacy adds to our understanding of the probability of events. Suppose we are playing a dice game and sometimes we use an electronic version connected to a source of radioactive decay (suitably calibrated to give a 1/6 chance of any given number) and sometimes we use a physical dice that we assume is essentially deterministic in its operation. In the first case, if we interpret quantum mechanics as having irreducible randomness, then there is a physical probability, in the second only epistemic. I cannot see how the cases differ in terms of the fairness or conduct of the game. From the human end-user perspective, there is only epistemic probability.I think we should distinguish between different types of epistemic probability. For want of better terms, let’s call subjective epistemic probability (SEP) the likelihood of an event X given the evidence available at time t, to some agent S. Let’s call objective epistemic probability (OEP) the likelihood of an event X given all the evidence that could in principle be available to S if S had the time and means to gather it.Now consider the dice example given earlier:Six people each choose a number on a dice. The dice is rolled and one person wins. Does the fact that the dice was, in fact, loaded make this competition unfair? No. Even if, because of the loading, it was a dead cert that the number 5 would win and the number 3 wouldn’t, there’s no unfairness – not even to the person who chose number 3. What matters is that everyone who enters enters with an equal epistemic (for-all-they-know) probability of winning.]Here the SEP for each person is 1/6 but the OEP is 1 for 5 and zero for all other numbers. What makes this fair on the face of it is that when people are choosing the numbers they all have the same SEP and so nobody is advantaged. Also, the fact that the organiser has a different pool of evidence does not seem to influence the likelihood of who wins. I would still say that this game is unethical though, because the organiser of the game does not have the same SEP as the participants in such a way that the organiser’s SEP better approximates the OEP and yet the participants are not informed of this.The game can be fair and yet unethical. But suppose the organiser knows that her nephew will participate and that his favourite number is 5. Is the game still fair? Or does our view of what people know when, influence our judgement of fairness?Was the RÂ£J case unfair as well as being unethical? I would say best practice would mean that the protocol for how the competition is actually run should be available to all in advance. So it was certainly unethical. The same is true of the winnowing case but now we have a problem, because it seems that the winnowing game is fair. But why? Is it unfair if calls are screened on the basis of odd versus even minutes yet fair if the organisers toss a coin on receipt of each call? I think it is fair in both cases. There is disparity between the SEP and OEP in the first but not the second case. But if we assume that people phone in at random with respect to odd and even minutes then at time t just before the game begins, even if the organisers know the objective epistemic probability for a participant selected at random, it is the same as their subjective epistemic probability. This is also true for any participant whilst the game is running prior to them initiating the call process. This is not true in the actual Richard and Judy quiz and so it was unfair.

4. My background is not philosophy so forgive me if I make any beginners mistakes.I actually don’t see how a dice can be deterministic? To me that suggests you know what you will get when you roll the die. I know there are a set of possible outcomes and that when I roll the die I will get one of them, but not which one? Is it deterministic because the set of outcomes are finite and known?I’m not sure I see the difference between physical and epistemic probability. Or perhaps I should say the difference between them in pragmatic terms – because as you say from a human perspective they are effectively the same thing?The idea of Subjective Probability and Objective Probability fits with my thinking; I would imaging there could be any number of Subjective Probabilities depending on number of observers and their perspective? While Objective Probability would be *absolute*? So would Subjective Probabilities be a function of the Objective? Would a fair game then be defined as when all participants have the same subjective probabilty? “Six people each choose a number on a dice. The dice is rolled and one person wins. Does the fact that the dice was, in fact, loaded make this competition unfair? No.”I don’t think the answer is as simple as no, I think it depends. Say there is a 7th person who owns the die and he does not participate in the game, but he knows number 5 will always win. Could you say this game was “subjectively fair”? But only as long as the owner of the die does not participate? But it would not be objectively fair? Or is this where physical and epistemic probabilities separate? The die has the physical probability of rolling a 5 100% all other 0?Although if you take {one of the} the definition{s} of fair – free from bias – then the game is never fair as the die is never free from bias. Although this equates game fairness to die fairness…”Is it unfair if calls are screened on the basis of odd versus even minutes yet fair if the organisers toss a coin on receipt of each call? I think it is fair in both cases.”I agree they would be both fair, but as I said only one of them is random. The coin technique would work regardless of whether the participants knew how it was being done and would not impact the behaviour of the participants; the other technqiue would only work if participants had no knowledge of it. However I think the important point is that all participants know that only 50% of calls are allowed through, if they don’t then it ceases to be fair because while all participants think they have the same subjective probability they actually do not.So with competitions could you define fair in human terms – if the behaviour of a participant would change given full knowledge a competition is defined as unfair? Effectively the organizers have to ensure that all participants have the same subjective probability?