Two-coin problem

Quote: kp

If you pull a coin from the box that has heads up, what is the chance that the other side will have heads? 2/3.

If you already have one of the two coins in your hand and it has heads up, what is the chance the other side is heads? 1/2.

I don't see where you've gained any additional information between these two examples. Either way, you're just looking at the heads side of one of two coins.

Quote: kp

My first answer was 2/3. Then I doodled on paper and came up with 2/3. Now a couple of days later I'm switching to 1/2.

If you pull a coin from the box that has heads up, what is the chance that the other side will have heads? 2/3.

If you already have one of the two coins in your hand and it has heads up, what is the chance the other side is heads? 1/2.

The first scenario is a compound event (parlay). The second scenario is an independent trial.

I feel like Darth Vader and im turning people to the dark side of 1/2 LOL!!!!!!!

Quote: vert1276

LMAO I know you guys think im totally messin with you but im not.

And I was just foolin' when I accused you of being tuttigym. But seriously, I'm having flashbacks. =)

Your points are good enough that I'm going back to examine the original question, but then again, I'm one of the one's who think picking the other door in Monty is pointless =P

Edit: Nevermind. 10 seconds and I think 2/3 too. Still wouldn't pick the other door though.

Quote: vert1276

1) I know there is at least one American flag stamp still in the box, correct?
2) I know the American flag stamp I am "observing" can not also be on the other side of the quarter, correct?
3) I still have 2 possible postage stamps that can be on the other side of the quarter, correct?

Now you all are saying there is a 66% chance the other side of the quarter has an American flag stamp on it. And Im saying there is only 2 possibility left now that I have observed one face and one of the 2 has to be the liberty bell stamp. please explain to me how there can be 3 possibilities left to be other the other side of the coin WITHOUT reaching back into the box?

Don't confuse possibility with probability. There are only two possibilities because there are only two images -- flag or bell. But the flag is twice as likely as the bell.

This has been suggested before (bird/fish/etc. analogy) but when you distinguish the sides it becomes easy, I hope. If you number the heads 1, 2, and 3, but you apply them randomly to the coins, you can either have
H1/H2, H3/T
or
H1/H3, H2/T
or
H1/T, H2/H3

Let's suppose I do that and put the coins into a bag, and I chose the arrangement randomly with equal probability from the above possibilities. You reach into the bag and pull out a coin with H1 showing. What are the chances that the flip side is either H2 *or* H3?

Quote: MathExtremist

Don't confuse possibility with probability. There are only two possibilities because there are only two images -- flag or bell. But the flag is twice as likely as the bell.

This has been suggested before (bird/fish/etc. analogy) but when you distinguish the sides it becomes easy, I hope. If you number the heads 1, 2, and 3, but you apply them randomly to the coins, you can either have
H1/H2, H3/T
or
H1/H3, H2/T
or
H1/T, H2/H3

Let's suppose I do that and put the coins into a bag, and I chose the arrangement randomly with equal probability from the above possibilities. You reach into the bag and pull out a coin with H1 showing. What are the chances that the flip side is either H2 *or* H3?

Said that Monday, too. We're just going to keep using the same arguments, so I'm with Mosca. The 4th graders that weren't convinced did the experiment and ended up with "Heads" around 65-70% of the time. If you don't want to do that, we're just going to be arguing in circles.

Maybe I'm oversimplifying things but....

I haven't read ANY of the responses since I was the first responder on page 1.

As I see it, past results have no influence on future actions.

You either pulled the two-headed coin out or not.

So it's a 50% chance the flip side is heads.

Apparently, I WAS oversimplifying things.

I started reading the thread. About half-way thru page 4, the fog cleared.

I then went back to page 2, to re-read this little comment in ChesterDog's Post:

Quote: ChesterDog

What's the probability of first seeing a randomly-selected coin as heads and then finding out that the coin is double-headed?

Yeah, phrased that way, it's obviously 2/3.

Quote: DJTeddyBear

Maybe I'm oversimplifying things but....

I haven't read ANY of the responses since I was the first responder on page 1.

As I see it, past results have no influence on future actions.

You either pulled the two-headed coin out or not.

So it's a 50% chance the flip side is heads.

If you pull a coin a million times, 25% of the time you will pull the normal coin on the tail side, 25% of the time you would pick it on the head side and 50% of the time you would get the other coin on head. These are your 4 possibilities:

Double headed coin, heads.
Double headed coin, heads.
Regular coin, heads.
Regular coin, tails.

Now let's take out the last possibility.
Double headed coin, heads.
Double headed coin, heads.
Regular coin, heads.
Regular coin, tails.

There are now 3 possibilities, two of which are that you picked the double headed coin.

HK -

Yeah, I know. I already did a follow-up.

And I've also come up with a simpler response:

There are three faces that are heads. You could have picked any one of them. Two of them have heads on the other face, therefore, 2/3 of the time it's the double headed coin.

Quote: cclub79

Said that Monday, too. We're just going to keep using the same arguments, so I'm with Mosca. The 4th graders that weren't convinced did the experiment and ended up with "Heads" around 65-70% of the time. If you don't want to do that, we're just going to be arguing in circles.

And since I don't like arguing in circles, I'll simply offer this:

I'll gladly offer anyone a wager of $100 to my $130 that *tails* will show when we play the game. If the odds of heads or tails are truly 1/2, that yields a healthy +15% EV. I'll play for as many trials as you want. Any takers?

(Of course, if I'm right, I have a +18% EV...)