Simple coin tossing problem?

[PrayingMantis]

This is a simple problem I thought of. Any help will be appriciated, because I'm a bit lost in it...

My friend has tossed a coin 4 times, and written down the results.

Now he tells me: "In this sequence, I got three heads, and one other result, which is either heads or tails. Guess what it was. I'll give you 100$ if you guess it right!"

Well, if I'm treating this in a non-conditional manner, the probability is of course 0.5 for each. So it makes no difference if I say heads or tails.

However, I know that for this 4 tosses sequence, the probability that there were only three heads, is 4:1 to the prob. there were four (i.e, there are 4 ways to get [3 heads, 1 tails], but only 1 to get [4 heads]). So it also makes some sense to assume that the answer is 0.8 for tails, 0.2 for heads. Therefore, I better say tails.

How should I solve this one?

[Ghazban]

Since each flip is totally indendent of all previous flips, the odds are still 50/50 for heads or tails even though 3 heads came on the previous 3 flips. The fact that its more likely to get 3 heads and 1 tails in 4 flips than 4 heads is totally irrelevant.

[TomCollins]

If he didn't tell you any information, and had you bet on 3 heads or 4 heads, then you are better off betting on 3 heads, since there are 4 times as many ways to pull this off.

But... he already told you the first results were heads. And I also bet on heads, because the previous results may indicate the coin is somehow deformed or shaped to land on heads more often.

Assuming a fair coin, its 50-50 for the 4th one.

[PrayingMantis]

[ QUOTE ]
But... he already told you the first results were heads

[/ QUOTE ]

The problem is that he didn't tell me that. He only told me "I got three heads, and one other result, which is either heads or tails". Nothing about first results and last result. Is it still 50-50?

And it's a fair coin.

[PrayingMantis]

[ QUOTE ]
even though 3 heads came on the previous 3 flips

[/ QUOTE ]

He didn't mention 3 heads came on the first three flips (i.e, on the previous 3 flips). He only said he got three "head" reasults, and one other result, that is either heads or tails. That could be in any order.

[kwyjibo00]

these guys are wrong. you should say tails. i'll let others elaborate.

[FloppedFlush]

It depends whether your friend was going to offer you this proposition regardless of what the results of the coin flipping was and what offer he was going to make in each situation. Say your friend went into the trial with the following rules:

if 4 heads - say 3 heads, what was the 4th?
if 3 heads - say 3 heads, what was the 4th?
if 2 heads - say 2 of whatever the first flip was, what was the 4th?
if 1 head - say 3 tails, what was the 4th?
if 0 heads - say 3 tails, what was the 4th?

then you've got the situation you describe in your post. Out of all the situations where your friend tells you he got 3 heads, the other flip being tails is much more likely than the other flip being heads.

On the other hand, say your friend had these rules:
if 4 heads - say 3 heads, what was the 4th?
if 3 heads - say 2 heads, what was the 4th?
if 2 heads - say 2 heads, what was the 4th?
if 1 head - say 2 tails, what was the 4th?
if 0 heads - say 3 tails, what was the 4th?

Now if he says he got 3 heads, the other flip is also a head 100% of the time.

Then there's the various stages in between to consider - when your friend's rules aren't quite so well defined (ie. say 3 heads 37% of the time and 2 heads 63% of the time, etc.).

So just like poker, it's at least as much about reading the player as it is about reading the cards.

[TomCollins]

Sorry for the misunderstanding. I took the 3 heads to mean 3 in a row.

The answer seems to depend on how your friend thinks.

So let's assume his method of telling you what he had was picking 3 of the 4 flips, and telling you those results. This will remove the element of trickery from your friend. It is 50-50 here (can you see why?).

If he decides to tell you what the majority was, you bet on tails, because there are 4x as many of these possibilities.

[Mano]

Definitely pick tails.

Possible outcomes (your probability space):

All 4 coins heads.
First coin tails, the rest heads.
Second coin tails, the rest heads.
Third coin tails, the rest heads.
Fourth coin tails, the rest heads.

All 5 of these outcomes is equally probable, so 80% (4/5) probability that there was a tail from information given.

[JTrout]

You've only got to figure out at what level your friend thinks.

Average guy thinks: " if I pose this question to my friend, he'll probably say tails."

guy thinking 1 higher level: "if I pose this question to my friend, he'll assume I'm trying to trick him- and will answer heads."

etc. etc.