Posted by on Oct 18, 2007 in Richard and Judy - and probability | 9 comments

# Channel 4 phone-in competition

“[T]he company behind a quiz on the Richard & Judy show was fined for urging callers to enter after winners had been picked.”

Regarding the upset over those who entered a Channel 4 phone-in competition (and were charged for doing so) when the winner had already been picked, I have to say, I don’t really get it.

You enter the competition with an epistemic probability of winning of, say, one in a million. You all know one million entered, and there will be only one winner. None of you have any idea who it will be.

Of course, your objective chance of winning (the physical probability, if you like) depends on how we set the parameters. For example, given the laws of nature and antecedent physical conditions, it may be that the winner was determined months or even years beforehand. Given these laws and initial conditions, your chances of winning are zero and those of Bert (the eventual winner) are 100%.

Does the fact that the physical probability of Bert winning is 100% mean the competition is not fair? Of course not.

But then why is it unfair if, unbeknownst to any punters, the winner is picked before the competition closes? Your epistemic chance of winning remains 1 in a million, even if you phone in after the winner is picked. Yes, objectively speaking, your objective probability of winning is then zero. But why does that make the competition unfair?

Unfortunately, probability theory is not really my forte. Anyone care to help me out?

[N.b. This case seems analogous to the following. 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.]

1. Seems to me it’s the timing thing. There is a way of determining who might win and who cannot – what time they call in. If suddenly a billion people called in after the winner had been decided, the probability of winning isn’t now a little under one in a billion, for the million or so that called first it’s still one in a million and the rest have zero.On the other hand http://www.digitalspy.co.uk/broadcasting/a46205/c4s-deal-facing-phone-in-investigation.html strikes me as a ludicrous basis for complaint.

3. I admit straight up that I am not particularly well versed in probability theory but it seems to me that if the physical probability of someone entering a competition is in fact 0%, assuming they are not aware of this, then that does seem unfair.The loaded dice is not a good analogy because all 6 of the ‘entrants’ had an equal chance of winning before choosing their number. To be analogous of the phone contest, imagine there are 12 people choosing a number on the dice. 6 of the entrants are allowed to choose a number before the dice is rolled and one of them wins. Then the other 6 would be allowed to choose after the contest was over. If only the first person to choose the correct number wins then it is difficult to see how this is could be considered fair. This could be gross misunderstanding on my part but what use is epistemic probability at the point where the physical probability is known?

4. If you phoned in late, you thought you had an epistemic chance of winning of one in a million, but in fact you turned out to have a chance of zero.Most people don’t, of course, have any understanding of the world as deterministic anyhow. But even if they did, there was a whole big antecedent condition that they “should have” known about but didn’t.And, to cap it all, they were charged for not-having a chance of winning.Tell you what, Stephen, if you send me a fiver (I take PayPal), I will enter you in a draw to win £100. There will be a maximum of 3 other entrants. Do you want any other information before taking up this probabilistically-attractive offer? If so, what, and why?If you don;t fancy it, I will change the rules; you don’t have to send me a fiver, you just have to post a comment here sayng “you’re on”. Does that any difference? If so, why?

5. Hi – just to clarify, epistemic probability is, roughly, the probability of P being true given the evidence available to you.So Potentilla’s first point – that if you (unknowingly) phoned in late, your epistemic probability of winning was zero, is not correct.However, I think P is right that it’s not just epistemic probability that’s relevant to fairness (if that’s the moral to be drawn from her example of the £5 bet to win £100).In fact, even when I posted this, I know the issue was not *just* epistemic probability. After all, if each of one hundred people thinks they have a 1/100 chance of winning, but in fact it’s a fraudulent competition and no one will actually win, then it’s unfair, whether or not the epistemic probability for each competitor is one in a hundred.That’s not to say epistemic probability is irrelevant, though. E.g. in a lottery, someone with inside knowledge of which number will win or is most likely to win does have an unfair advantage.I’m still puzzled.In fact, Ron has pointed out additional factors (e.g. they call after the official cut off, that complicate the case still further).

6. Stephen – W – the event that I winE – the evidence available (laws of nature, rules of the competition etc)What you refer to as objective probability is P(W)What you refer to as epistemic probability is P(W|E).Note that they are different probabilities and can (in general) take on different values.The unfairness arises when the competition organiser uses a different E to me and doesn’t disclose it. So I am unable to ‘price’ the bet correctly because I’ve been misled.In the dice example, it’s reasonable to assume that the dice is fair as part of my E. If not (and the competition organiser *does know* this as part of his E), I’m being conned.In the phone example, it’s reasonable to assume that I have a 1/N chance of winning, if N people actually phone in. Again, if the organiser chose the winner six weeks ago, it’s unfair.So, to answer your very last question (which probabilities are relevant): they both are.snafu.

7. Sorry, re e.p.I’m not sure what moral is to be drawn either, hence the reason I phrased it as a thought experiment.I think maybe it’s the case that people feel the epistemic probability in not relevant at all without the condition being met that the organisers have disclosed all the relevant information that they “knew or ought to have known” (phrase from English case lw, although I can’t remember exactly where).

8. I think the problem is basically this: the company this was withholding information such that the population thought they could win, when in fact they could not. In this way, they encouraged the population to become contestants and to spend money which they otherwise would not have done. In the loaded dice scenario, the company also withholds information. But here this withholding of information does not alter the contestant’s odds of winning. With or without the dice being loaded, the contestant still has a one in six chance of guessing the right number. As such, I agree that it isn’t unfair.(Aside: our sense of morality here seems tied to whether or not the contestants suffer (i.e. pay money), and perhaps whether or not the company/casino profits, from this manipulation. The manipulation itself, though conscious in both cases, is apparently not what we’re judging. No quibbles, but interesting.)

9. Ho ho, Potentilla.I have an even better game! It’s already in progress, but you can join if you want. Everyone send me a fiver by Paypal, and next week I’ll pick a winner who gets half. A friend of mine has already entered. (But watch out. I may just close the competition and keep your money. I may have already done so.)