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Hi, you have probably heard of this classic math question about coin flipping, but in case you haven't I will tell it to you.
Your friend tosses two coins, then asks you to guess how the coins landed. You reply that you cannot know. Then your friend reveals that one of the coins he threw landed heads. Now how did the other coin land?
(Correct) answer: There is a 67% chance that the other coin landed tails. Why? Because after he told you one was heads, the remaining possibilities of the coin lands were heads-heads, heads-tails and tails-heads, therefore in two out of three cases the other coin is tails. If he told you that the first coin thrown was heads, then the chances would be 50-50, since there would be only two possibilities, heads-heads and heads-tails.
My opinion: WHAT? That is the stupidest thing I have ever heard. Two coins were thrown, coins don't have a memory, no matter what your friend tells you, the chances of other coin will always be 50-50. Let's look at this from a different perspective.
We all know that if two coins are thrown, there is only 50% chance that both land same side up. But according to this, after he tells you one coin landed X, there is 67% chance that the other coin landed different side. Do we really think that if someone tells you something, the probabilities of coin flips change?
Let's create a fictional scene, where you use the (correct) math. You walk down the street, there is this guy who tells you he will give you $5 if you can throw two heads in a row. You think ok, 25% chance. Then he tells you it gets better, one of the coins you will throw is heads on both sides. Now you think wow, you throw that first and 50% chance you get second one heads as well. But then he says to you he won't tell do you throw the one sided coin first or second. Now you feel bummered since you only have 33% chance since the three possible ways the throwing can go are heads-heads, tails-heads and heads-tails.
Exactly, that does not make any sense what so ever. It does not matter when you or he, or anyone throws the one sided coin, you will always have 50% chance with other coin. Throwing two coins always has a 50% chance that both coins land same side, even if you know how one of the coins landed. FACT!
 
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Calgary25951 Posts
Random:
HH x
HT o
TH o
TT x
50%
After knowing at least one was heads:
HH x
HT o
TH o
TT
66.6%
O_o
I think you're just thinking about it wrong. His knowledge isn't affecting the outcome - the outcome was random but he gave you specific information about the result that eliminates one of the possibilities.
If he flipped a coin, told you it was heads, and then flipped the other coin, there would be a 50% chance one of them was tails. But that's not the same thing. In scenario 1, TT was an option was that later eliminated with information. In scenario 2, TT is never a possibility so it doesn't factor in.
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edit again - the last part of chill's post makes the most sense out of this lol
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It's the classic Montey's door scenario or what not. Learn it in every intro probability course. It's actually not intuitive if you haven't studied probability yet, so it's understandable how people get this wrong all the time.
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On July 16 2011 02:38 Chill wrote:
Random:
HH x
HT o
TH o
TT x
50%
After knowing at least one was heads:
HH x
HT o
TH o
TT
66.6%
O_o
I think you're just thinking about it wrong. His knowledge isn't affecting the outcome - the outcome was random but he gave you specific information about the result that eliminates one of the possibilities.
It eliminates one of the possibilites, but it also make the heads-heads possibility twise as possible as before making it 50-50.
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On July 16 2011 02:35 Sea_Food wrote:
Let's create a fictional scene, where you use the (correct) math. You walk down the street, there is this guy who tells you he will give you $5 if you can throw two heads in a row. You think ok, 25% chance. Then he tells you it gets better, one of the coins you will throw is heads on both sides. Now you think wow, you throw that first and 50% chance you get second one heads as well. But then he says to you he won't tell do you throw the one sided coin first or second. Now you feel bummered since you only have 33% chance since the three possible ways the throwing can go are heads-heads, tails-heads and heads-tails.
In this scenario, the outcomes you listed do not all have equal probability, hence the error in your calculations.
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On July 16 2011 02:40 lvatural wrote:
It's the classic Montey's door scenario or what not. Learn it in every intro probability course. It's actually not intuitive if you haven't studied probability yet, so it's understandable how people get this wrong all the time.
I have studied probability, and from a mathbook, I first saw this question, but I did not want to argue this with math teacher.
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On July 16 2011 02:43 Jumbled wrote:Show nested quote +On July 16 2011 02:35 Sea_Food wrote:
Let's create a fictional scene, where you use the (correct) math. You walk down the street, there is this guy who tells you he will give you $5 if you can throw two heads in a row. You think ok, 25% chance. Then he tells you it gets better, one of the coins you will throw is heads on both sides. Now you think wow, you throw that first and 50% chance you get second one heads as well. But then he says to you he won't tell do you throw the one sided coin first or second. Now you feel bummered since you only have 33% chance since the three possible ways the throwing can go are heads-heads, tails-heads and heads-tails.
In this scenario, the outcomes you listed do not all have equal probability, hence the error in your calculations.
Then why do all the outcomes in the orriginal story have equal probabilities?
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Calgary25951 Posts
On July 16 2011 02:42 Sea_Food wrote:Show nested quote +On July 16 2011 02:38 Chill wrote:
Random:
HH x
HT o
TH o
TT x
50%
After knowing at least one was heads:
HH x
HT o
TH o
TT
66.6%
O_o
I think you're just thinking about it wrong. His knowledge isn't affecting the outcome - the outcome was random but he gave you specific information about the result that eliminates one of the possibilities. It eliminates one of the possibilites, but it also make the heads-heads possibility twise as possible as before making it 50-50.
He told you the information after flipping the coins. It isn't twice as likely.
Think about it:
Scenario 1: I flip 4 coins. I tell you 3 of them are heads. Are you saying the chance of the last one being heads is the same as it being tails? Obviously not:
HHHH
THHH, HTHH, HHTH, HHHT
There are way more possibilities to have 1 tails and 3 heads than all 4 heads.
Scenario 2: I flip 3 coins. They are all heads. What is the chance of the fourth coin being heads? 50%. Coins don't have memories.
HHHH
HHHT
These are the only possibilities.
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On July 16 2011 02:39 rawb wrote:
edit again - the last part of chill's post makes the most sense out of this lol
-.-
The thing chill said in last part was also in the OP.
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Well, I don't really understand completely either, but I think the confusion results from one focusing on single coin instead of two coins which were thrown overall. He's not asking you the probability of 'a' coin landing tails, but what the result of the coin that is not head is.
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you have the question slightly wrong, as the coins aren't interdependent, it is still a 50% chance.
i believe the actual problem you are referring to involves 3 doors, two bad prizes, and one good prize.
you pick one door, from which you will receive the prize. the host then shows you one of other the doors with a bad prize, and asks you if you would like to switch your choice of door. you should, because:
initially you had a 33% chance of getting the good prize, meaning there is a 66% chance that the good prize is behind the other two doors. After the host shows you one of the other doors had a bad prize, the percentages remain the same, and your current door has a 33% chance, while the other one left has a 66% chance, meaning if you want the good prize, switch doors.
i hope this was clear to everyone.
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Calgary25951 Posts
On July 16 2011 02:46 Sea_Food wrote:Show nested quote +On July 16 2011 02:39 rawb wrote:
edit again - the last part of chill's post makes the most sense out of this lol -.- The thing chill said in last part was also in the OP.
I like how you're coming into this as if a famous example of probability is wrong and you are right. Maybe you should try to understand why you're wrong 
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Edit: misread his hypo. It's actually pretty confusing lol
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Calgary25951 Posts
On July 16 2011 02:46 Hesmyrr wrote:
Well, I don't really understand completely either, but I think the confusion results from one focusing on single coin instead of two coins which were thrown overall. He's not asking you the probability of 'a' coin landing tails, but what the result of the coin that is not head is.
Very good point. In Scenario 1, only one of the coins is heads. It doesn't matter which one. In Scenario 2, only the first coin can be heads. That's why Scenario 1 is more likely.
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You understand that one coin doesn't affect the outcome of the other coin, which is perfectly true. But the probability of 2/3 that you give IS affected by the result of the first coin.
The first coin doesn't affect the result of the second coin, but does affect the result when both coins are considered together, which is what you're doing.
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So COIN 1 have 50%/50% chance of getting head & tails.
COIN 2 also have 50%/50% chance of getting head & tails.
Thus
TT
TH
HT
HH
is equally probable. Now one coin landed heads, so visualizing the information:
(Bolded is the coin your friend said was heads.)
TT
TH
HT
HH
Two T, one H. I bolded the other coin with red to emphasize the forementioned result have not come from 'single' coin.
Edit: Just addressing this because I think this is more clearer.
Then why do all the outcomes in the orriginal story have equal probabilities?
TT 25%
TH 25%
HT 25%
HH 25%
The possibility of other coin being H - meaning 2 coins flipped heads - do not change from 25% to 50% because the coins indeed do flip TT 25% of the times. We are not just excluding the TT scenario because we have new information that at least one coin landed H.
So instead of thinking 2/3, one could also say there is 50% chance that coins will flip HT and 25% chance that coins will flip HH.
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I think i understand the mixup. if you have TH and HT as two separate options, shouldn't there also be two HH's? as your double-headed coin could land on either head.
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On July 16 2011 02:35 Sea_Food wrote:
We all know that if two coins are thrown, there is only 50% chance that both land same side up. But according to this, after he tells you one coin landed X, there is 67% chance that the other coin landed different side.
Can someone try to explain me this?
Because acording to the correct math, if my friend tosses two coins 1 000 000 times, and tells me how one of the coins landed each time, I will say each time that the chacnes coins landed different sides is 67%, which means if im rigth about 677 777 times the coins did land different side.
Now if he didnt tell me anything, the chances would be the coins landed different side only about 500 000 times
On July 16 2011 02:47 Chill wrote:Show nested quote +On July 16 2011 02:46 Sea_Food wrote:On July 16 2011 02:39 rawb wrote:
edit again - the last part of chill's post makes the most sense out of this lol -.- The thing chill said in last part was also in the OP. I like how you're coming into this as if a famous example of probability is wrong and you are right. Maybe you should try to understand why you're wrong
If I didnt think that a famous example of probability is wrong and I are right, I would have not wrote this blog.
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Yeah, this is sort of like the classic Monty Hall problem, which has been repeated seemingly 32834720487 times...
I hope you get it; it's pretty interesting 
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EPT
