brain teasers #1

[bugzpodder]

I have a bunch of brain teasers, this will be #1 of a bunch of them. personally i think they are very interesting.

A box contains two coins. One coin is heads on both sides and the other is heads on one side and tails on the other. One coin is selected from the box at random and the face of one side is observed. If the face is heads, what is the probability that the other side is heads?

[snakeeyes1000]

1/2
(50%)
Is this simple, or am I missing something and made myself an idiot?

[Gandalf_Grey_]

I am pretty sure that there was a similar riddle posed a while ago involving a game show and goats . Let me think about it.

[Gandalf_Grey_]

ok i have thought about it. The answer is the probability that you have the two headed coin in your hand is 2/3 and the probability that you have the regular coin in your hand is 1/3. The reason for this is because it is assumed that you have a head facing you when you pick up the coin. if you don't then you must put it back and try again (or something like that) if you could have a tail facing you the odds would be 50/50 but since if you get that 1 in 4 chance that you have a tail facing you it stacks the odds. If you would like i made a program in VB to demonstrate this (it is horribly innefficiant i know but i did it quickly with no thought of design really.

Private Sub Form_Click()

Dim HypoTrue As Double
Dim HypoFalse As Double

Dim Random1 As Byte
Dim Random2 As Byte

Dim i As Double
Dim MAX As Double

MAX = 1000000

For i = 1 To MAX

Coin(0, 0) = True
Coin(0, 1) = True 'true = heads

Coin(1, 0) = True
Coin(1, 1) = False ''false = tails

Do

    Random1 = Int(Rnd * 2) 'which coin
    Random2 = Int(Rnd * 2) 'which side

Loop While Coin(Random1, Random2) <> True 'if you draw a coin with a tail

If Random1 = 0 Then 'if you drew the double headed coin
    HypoTrue = HypoTrue + 1
    
End If

If Random1 = 1 Then 'if you drew the normal coin
    HypoFalse = HypoFalse + 1

End If
    
Next i

MsgBox "The hypothesis was right " & 100 * (HypoTrue / MAX) & " of the time, and wrong " & 100 * (HypoFalse / MAX) & " of the time"
End Sub

[snakeeyes1000]

if you randomly pull one out, and the side you look at is heads, I promise it doesn't matter. Either it is the double-headed one or it has a tail on the other side. It's 1/2. If the question was "what is the probability that the other side of the coin you took is heads" then you would be right. sorry.

[bugzpodder]

yeah i am sorry, the question should be "whats the probability that the other side is heads". thx snakeeyes my fault

i have edited my first post, but here is the *original* question again: A box contains two coins. One coin is heads on both sides and the other is heads on one side and tails on the other. One coin is selected from the box at random and the face of one side is observed. If the face is heads, what is the probability that the other side is heads?

So whats your final answer?

[Gandalf_Grey_]

Originally posted by snakeeyes1000
if you randomly pull one out, and the side you look at is heads, I promise it doesn't matter. Either it is the double-headed one or it has a tail on the other side. It's 1/2. If the question was "what is the probability that the other side of the coin you took is heads" then you would be right. sorry.

the stipulation in the question which makes it not 50/50 is the fact that the one you are looking at must be heads. there is a 3 in 4 probability that you drew a heads out of the box however if it is tails it is assumed you put it back and try again. I proved this problem by writeing a small program which did it a very large number of times. Please find the error in my code

[Gandalf_Grey_]

my final answer is 2/3 of the time you are looking at the coin which has 2 heads

[nishantp]

But Gandalf, the question is only adressing a certain situation. You would be right if the question was just:

what is the probability that the other side is heads?

But the question is:

If the face is heads, what is the probability that the other side is heads?

The question is only addressing a situation in which you are already looking at heads. In that case, you either have the normal coin, or the coin with only heads. That makes the odds 50%.

IMO

[Gandalf_Grey_]

oh **** ya. nevermind. I guess i likened it to the goat and car question.

[bugzpodder]

The correct answer is 2/3
here is 2 links for solutions and some interesting follow ups: http://www.thewizardofodds.com/math/prob16s.htm
http://maa.truman.edu/solutions.html#sol4

[Guv]

Why is this causing controversy? A straight forward analysis is obvious. Instead of using probability formulae whihc are not always convining, consider what you would expect to happen in 100 trials.

Two coins have a total of 4 sides. If you did this experiment 100 times, each side would be seen about 25 times. About 75 times you would see a head and about 25 time you would see a tail.

Of the 75 times you saw a head, about 50 times you would be looking at a head from the two headed coin.

Ergo: Probability is about 50/75 or 2/3 that the other side is also a head.

[snakeeyes1000]

the way bug's original problem was worded, before it was edited, was about the probability of getting the two headed coin, which would of course be 1/2.

[Guv]

the way bug's original problem was worded, before it was edited, was about the probability of getting the two headed coin, which would of course be 1/2.

How was the problem worded to cause the probability to be 1/2?

Was the coin picked without looking at either face? The following wording would still result in the probability of the two headed coin being 2/3.

A box contains two coins. One coin is heads on both sides and the other is heads on one side and tails on the other. One coin is selected from the box at random and the face of one side is observed. If the face is heads, what is the probability that the the two headed coin was picked?

The answer to the above is 2/3. The answer to the following is 1/2

A box contains two coins. One coin is heads on both sides and the other is heads on one side and tails on the other. One coin is selected from the box at random and neither face is examined. What is the probability that the two headed coin was picked?

[snakeeyes1000]

actually the question was "If the side you're looking at is heads, what is the probability that you have the two-headed coin? 50% of course.

[Guv]

SnakeEyes1000: Once you have information about the coin you picked, the probabilities change. The following is dead wrong.

actually the question was "If the side you're looking at is heads, what is the probability that you have the two-headed coin? 50% of course.

Before you look at the coin the probability is ½ for each coin. Once you look at one side, the probabilities change.

Suppose you picked a coin at random and saw a tail on one side. Now what is the probability of having picked the two headed coin? Obviously it is zero. The information has changed the probability from ½ to zero. Similarly, seeing a head changes the probability from ½ to 2/3. The latter change in probability is a bit harder to calculate and is not as obvious as the former change, but it is just as valid.

[bugzpodder]

That is a very interesting analysis.

[snakeeyes1000]

he said orignally to keep drawing until you get heads side up, fool If you followed his instructions, you got 50/50. If you didn't see the orignal question, what is the probability you will successfully argue about it with me?

[bugzpodder]

The original question asks if you see heads, whats the probability that it's the two heads coin. what i don't get is -- right now the question asks what is the probability that the other side is heads -- isn't it the same thing? if the other head is heads then its the two heads coin. if its not then its not. it goes the other way too. if its the two heads coin the other head is heads. otherwise its not.

[snakeeyes1000]

Let's make the side pool.

H
H
H
T

If you see heads, you know at least one of the 3 heads is NOT on your coin, so that's two heads accounted for. We'll take them out.

H
T

Both answers are 50/50. Using common sense, this probability question is the same as the other one, you are a smart guy.

[snakeeyes1000]

Let's make the side pool.

H
H
H
T

If you see heads, you know at least one of the 3 heads is NOT on your coin, so that's two heads accounted for. We'll take them out.

H
T

Both answers are 50/50. Using common sense, this probability question is the same as the other one, you are a smart guy. They are both really the same question. You had me fooled

[Guv]

SnakeEyes1000: Does your last post make sense to you?

How about the following analysis?

When you consider the following possibilities, it looks to me like 2/3 of the heads are on the two-headed coin. Hence, when you see a head the probability that it came from the two-headed coin is 2/3.

H From two headed coin
H From two headed coin
H From normal coin
T From normal coin

[bugzpodder]

Guv's analysis makes much more sense to me
snakeeyes is saying: since both coin has heads, so what you see (heads) doesn't matter, it's still be 50/50. unfortunately thats incorrect. let me ask you, snakeeyes, now that you know its heads, which coin do you think it is more likely? the one with two heads or the one with 1 head?? (probability of 2 headed coin vs probability of normal coin)

if u r still not convinced, lets use your analysis on 100 coins, 99 with 2 heads and 1 normal one

you have
199 H, 1T
since you see H, and also, you know that at least 197 H is not your coin (hehe ), so that's 198 heads accounted for. We'll take them out.
H
T
which makes the coin you pick up 50% as likely to be the normal coin. Both answers are 50/50 (this is pretty good, snakeeyes ).

do you think thats correct?