Some more probability thoughts

[peterbirks]

Weirdly, it isn't the counterintuitive 13/27 answer that caused the controversy (thus showing that you guys actually do understand numbers), but whether the starting point is a half or a third (thus showing that we don't really understand semantics, language and philosophy).

I've been thinking about this "two children" problem, and I think I've come up with an interesting, and probably solvable (given a few assumptions about certain likelihoods) poser.


Let's apply this to gambling!

A man walks in. Slaps his palm down on the table on top of two coins and looks at them. He says to you (Mr Aardvark) "There are two coins here. At least one of them is a head."

A second man (a Mr Birks) (who knows no more than you do about this state of affairs) now says: "I will offer you 3-to-2 on both coins being heads".

Do you take the bet?

Well, you, Mr Aardvark, are in the 50:50 camp. So obviously you do.

As it happens, Mr Birks wins the bet (the second coin is a tail), and you are ten quid down.

"Mr Aardvark", scoffs Mr Birks, "You are a fool! Everyone knows that the chance of the other coin being a head is one-third!"

But you, Mr Aardvark, are not non-plussed. "Double or nothing?" you say.

"Hah HAH!" says Mr Birks. "This is going to pay for my holiday!"

The first man then walks out of the door. Ten seconds later he comes back in. He repeats the exercise, and says:

"At least one of these coins is a tail."

At this point, Mr Birks (foolishly) says "ahh, it's exactly the same problem, so clearly the chance of the second coin being a tail is still 1/3." He offers you 3-to-2, and you absolutely lump on, because now you are certain (incorrectly, as it happens, but we can cover that later) that the chance of the other coin being a tail is 50%.

Let's take the sample of four.

(A) Coins are HH. Man says "At least one of these coins is a head"
(B) Coins are TT. Man says "At least one of these coins is a tail"
(C) Coins are TH. Man says "At least one of these coins is a tail"
(D) Coins are TH. Man says "At least one of these coins is a head"

The "law of restricted choice" in A and B causes the probability of the other coin to be the same to move up to 50% from 33%.



But, suppose the man comes in the second time, tosses the two coins and puts his palm down, and, once again, says... "At least one of the coins is a head". Once again, Mr Birks offers you 3-to-2 on the other coin being a head. Do you, Mr Aardvark, still lump on?

This is a tougher prospect.

Clearly, therefore, what matters here (in terms of making money) is the sample of "choices of what to say" held by the man who tosses the coin.

Suppose he comes in, tosses the two coins, looks at them. Says nothing. And then tosses the two coins again. At which point he says "At least one of the coins is a head".

Even you, Mr Aardvark, will deduce from the fact that the man said nothing after the first toss that there is a state of affairs that commits him to silence. Thus, when he makes the second toss and states "At least one of them is a head", you will induce (perhaps incorrectly, but it's a reasonable induction) that his previous silence was because he had tossed two tails.

At this point you, Mr Aardvark, will conclude that the chance of the other coin being a head is 1/3.

In real life, of course, Mr Birks would deduce as soon as the man said the second time "at least one of them is a tail" (having said the first time that "at least one of them is a head", that the chance of the other coin being a tail was, because of the law of restricted choice, 50%, not 33.3%.


The key, therefore, is what the man is instructed to say. Even the above is simplistic.

I have, in fact, assumed two possibilities (nos (1) & (4) below) when there are more than this. remember, we are toalking about real prop bets here.

1) The man is told (or has decided) to say either "at least one is a head" or "at least one is a tail". If it's a head and a tail, he can say either.
2) The man is instructed to say "at least one is a tail" ONLY if both are tails. Otherwise, he says "at least one is a head".
3) The man is instructed to say "at least one is a head" ONLY if both are heads. Otherwise, he says "at least one is a tail".
4) The man is instructed to say "at least one is a head" if at least one is a head. Otherwise, he stays silent and tosses again.
5) The man is instructed to say "at least one is a tail" if at least one is a tail. Otherwise, he tosses again.

There are many other combinational possibililties, and clearly it would be erroneous to apply an equal probability to all of them. So let's just stick to casees (1) and (4) for the moment.


Now, let's go back to the first coin toss again. He tosses it, and says "At least one is a head". What is the chance that the other is a head?

If we take the simple sample here (choices of (1) and (4) being the "actual state of affairs") , we could say: "He might have been told to stay silent if neither was a head, or he might have been told to say either "at least one is a head" or "at least one is a tail" (his choice) if it was a head and a tail. These are equally likely states of affairs (under the principle of 'in the beginning, everything was even money')".

Now, this is where it gets interesting. In effect, if we are talking about having a bet on this, we know that there's an evens chance that it's 33.3% (state of affairs 4 is true), and an evens chance that it's 50:50 (state of affairs 1 is true).

This gives us the intermediate real, I'm prepared to put my money on it, chance of 41.67% that the other coin is a head!

Once you add in the other "posible states of affairs", things become more complicated. I can return to that later.

Now, if the man goes out, comes back in, and says "At least one is a tail", I think that the odds shift directly to 50%.

However, if he once again says "at least one is a head", we have an interesting calculation.

It's my guess (and it is a guess -- I haven't seriously worked it out) that this would shift the best "prop bet" likelihood to about 37.75% that the other coin is a head.

For every time the man comes in and says "at least one is a head", rather than saying "at lest one is a tail" or saying nothing and tossing again, you halve the distance between your previous percentage and 33.33%.

If anyone wants to test this financially, I am happy to walk into a room and be the coin tosser and statement maker while two people bet against each other!.


PJ

Comments

[real-aardvark.livejournal.com]

(Er, not quite on the last para. On the alternative view that villain knows you're looking for a pair, but doesn't know which one, then A and B coalesce on the assumption that 'o' is not a pair. In that case your odds are 33/66, because villain actually has more information, implicitly in the cards.)