"[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.]

## Comments

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.

One of the parameters of your 'objective chance' or 'physical probability' is whether you phone in or not. All other parameters being equal, if you phone in, in-time, along with others, making 1 million entrants, then your chance is 1 in one million. If you don't phone in, your chance is zero. The unfairness on the part of this contest is that by closing it before your call, but at the same time accepting your call, the program is artificially treating you as if you didn't phone in - they are forcing your chance to be zero.

However, with a sufficient capacity system to take and record all entered calls in a finite time, and with a pre-determined and precise cut-off time, then all calls received before cut-off are in the pot, and those after are not. That seems fair.

But then comes the actual unfairness for which too-late entrants have a just complaint. Though they may have been excluded from the contest fairly, they are being charged premium rate for calls that cannot increase there chance above zero - having been excluded. This might be because the cut-off is not precise, the phone system is queuing and charging them before omitting them from the contest, they unknowingly phone too late, etc.

To those included the 'cost' of 'entering' is the premium rate cost of the call. To the failed caller this is the 'cost' of 'attempting to enter'. Having said that, as long as the potential to be excluded yet charged is clearly stated any caller is knowingly accepting those conditions. I'm sure many callers call thinking they have called in-time - i.e. before the 'lines are now closed' prompt appears on the screen - when in fact they are connected and put in a queue.

Given the limited capacity of the system and its queuing techniques it could be made more fair:

1) Cut-off as precisely as possible.

2) Anyone in the call queue (i.e. already connected) charge at a lower rate, or zero. This will depend on the costs to the service providers in the chain - programme, channel, contest service providers, telephone service providers, etc.

3) Stop accepting new calls prior to the cut-off. It surely isn't beyond the capability of communications engineers, who understand queuing theory very well, to calculate a call processing rate and determine a call rejection time before contest cut-off time, based on the current queue size and the queue processing rate.

4) Don't use a queing system. Let all calls ring unconnected, until actually connected and accepted for the competition. When the 'lines are closed' really close the lines and present an unobtainable tone.

I suspect the competition companies do understand queuing theory extremely well, and rely on this for the shed-loads of money they are making off gullible callers who are charged before and after the phone lines are 'closed'. And as for option 4, think of the revenue lost if the callers' calls are not accepted and queued!

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?

thoughtyou 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?

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).

W - the event that I win

E - 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.

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 allwithout 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).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.)

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.)