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Posted: 4/24/2013 4:23:35 PM EDT
[Last Edit: 4/24/2013 4:54:22 PM EDT by XterraJohn]
On the game show Deal or No Deal, where there's 26 cases that each contain a certain dollar amount and only one of them contains $1,000,000:
Once you've opened all of the cases and you're down to two cases left, yours and the one on the stage, and you know that one of them has $1,000,000 and the other has $0.01 and they offer you the chance to trade cases, statistically, should you change cases or not? I contend that at that point you have a 50/50 chance of your case containing $1,000,000, while my friend contends that there's a 1/26 chance of that, while if you were to trade cases, you would then have a 50/50 chance. Who is wrong? Edit: spelling, changed 1/50 to 1/26. 


You are incorrect.


"During the second 100 days, we will design, build and open a library dedicated to my first 100 days." Barack Obama, May 9 2009

Originally Posted By XterraJohn:
On the game show Deal or No Deal, where there's 26 cases that each contain a certain dollar amount and only one of them contains $1,000,000: Once you've opened all of the cases and you're down to two cases left, yours and the one on the stage, and you know that one of them has $1,000,000 and the other has $0.01 and they offer you the chance to trade cases, statistically, should you change cases or not? I content that at that point you have a 50/50 chance of your case containing $1,000,000, while my friend contends that there's a 1/50 chance of that, while if you were to trade cases, you would then have a 50/50 chance. Who is wrong? Your friend. Two cases, P = 0.5, the other 24 are irrelevant. 

Ever tighter grows the noose around the neck of the lawabiding.

One in fifty percent.
Your friend is correct. 


Originally Posted By Him:
Originally Posted By XterraJohn:
On the game show Deal or No Deal, where there's 26 cases that each contain a certain dollar amount and only one of them contains $1,000,000: Once you've opened all of the cases and you're down to two cases left, yours and the one on the stage, and you know that one of them has $1,000,000 and the other has $0.01 and they offer you the chance to trade cases, statistically, should you change cases or not? I content that at that point you have a 50/50 chance of your case containing $1,000,000, while my friend contends that there's a 1/50 chance of that, while if you were to trade cases, you would then have a 50/50 chance. Who is wrong? Your friend. Two cases, P = 0.5, the other 24 are irrelevant. This. 

Tree hugging Gun totin' Free love Capitalist Hippie
Yeah, that's how I roll. Do you hear the people sing.... 
Originally Posted By HRomberg:
Originally Posted By Him:
Originally Posted By XterraJohn:
On the game show Deal or No Deal, where there's 26 cases that each contain a certain dollar amount and only one of them contains $1,000,000: Once you've opened all of the cases and you're down to two cases left, yours and the one on the stage, and you know that one of them has $1,000,000 and the other has $0.01 and they offer you the chance to trade cases, statistically, should you change cases or not? I content that at that point you have a 50/50 chance of your case containing $1,000,000, while my friend contends that there's a 1/50 chance of that, while if you were to trade cases, you would then have a 50/50 chance. Who is wrong? Your friend. Two cases, P = 0.5, the other 24 are irrelevant. This. Right, if you traded cases you are at 5050. I am assuming you traded places. The parameters stay the same unless you change the original parameters. 


Originally Posted By CaptPlanet:
Originally Posted By Him:
Originally Posted By XterraJohn:
On the game show Deal or No Deal, where there's 26 cases that each contain a certain dollar amount and only one of them contains $1,000,000: Once you've opened all of the cases and you're down to two cases left, yours and the one on the stage, and you know that one of them has $1,000,000 and the other has $0.01 and they offer you the chance to trade cases, statistically, should you change cases or not? I content that at that point you have a 50/50 chance of your case containing $1,000,000, while my friend contends that there's a 1/50 chance of that, while if you were to trade cases, you would then have a 50/50 chance. Who is wrong? Your friend. Nah. You're wrong. Your friend is right. This is kind of still wrong, but shows it fine: http://www.youtube.com/watch?v=Zr_xWfThjJ0 No comprendo. Explain please. 

Tree hugging Gun totin' Free love Capitalist Hippie
Yeah, that's how I roll. Do you hear the people sing.... 
Originally Posted By HRomberg:
Originally Posted By CaptPlanet:
Originally Posted By Him:
Originally Posted By XterraJohn:
On the game show Deal or No Deal, where there's 26 cases that each contain a certain dollar amount and only one of them contains $1,000,000: Once you've opened all of the cases and you're down to two cases left, yours and the one on the stage, and you know that one of them has $1,000,000 and the other has $0.01 and they offer you the chance to trade cases, statistically, should you change cases or not? I content that at that point you have a 50/50 chance of your case containing $1,000,000, while my friend contends that there's a 1/50 chance of that, while if you were to trade cases, you would then have a 50/50 chance. Who is wrong? Your friend. Nah. You're wrong. Your friend is right. This is kind of still wrong, but shows it fine: http://www.youtube.com/watch?v=Zr_xWfThjJ0 No comprendo. Explain please. Nah, the parameters stay the same unless you change the parameters. 


Originally Posted By HRomberg:
Originally Posted By CaptPlanet:
Originally Posted By Him:
Originally Posted By XterraJohn:
On the game show Deal or No Deal, where there's 26 cases that each contain a certain dollar amount and only one of them contains $1,000,000: Once you've opened all of the cases and you're down to two cases left, yours and the one on the stage, and you know that one of them has $1,000,000 and the other has $0.01 and they offer you the chance to trade cases, statistically, should you change cases or not? I content that at that point you have a 50/50 chance of your case containing $1,000,000, while my friend contends that there's a 1/50 chance of that, while if you were to trade cases, you would then have a 50/50 chance. Who is wrong? Your friend. Nah. You're wrong. Your friend is right. This is kind of still wrong, but shows it fine: http://www.youtube.com/watch?v=Zr_xWfThjJ0 No comprendo. Explain please. 


Guys, this is a trick question.



Originally Posted By HRomberg: Originally Posted By CaptPlanet: Originally Posted By Him: Originally Posted By XterraJohn: On the game show Deal or No Deal, where there's 26 cases that each contain a certain dollar amount and only one of them contains $1,000,000: Once you've opened all of the cases and you're down to two cases left, yours and the one on the stage, and you know that one of them has $1,000,000 and the other has $0.01 and they offer you the chance to trade cases, statistically, should you change cases or not? I content that at that point you have a 50/50 chance of your case containing $1,000,000, while my friend contends that there's a 1/50 chance of that, while if you were to trade cases, you would then have a 50/50 chance. Who is wrong? Your friend. Nah. You're wrong. Your friend is right. This is kind of still wrong, but shows it fine: http://www.youtube.com/watch?v=Zr_xWfThjJ0 No comprendo. Explain please. 1/2 > 1/26 

"During the second 100 days, we will design, build and open a library dedicated to my first 100 days." Barack Obama, May 9 2009

Originally Posted By AR4U:
Originally Posted By HRomberg:
Originally Posted By CaptPlanet:
Originally Posted By Him:
Originally Posted By XterraJohn:
On the game show Deal or No Deal, where there's 26 cases that each contain a certain dollar amount and only one of them contains $1,000,000: Once you've opened all of the cases and you're down to two cases left, yours and the one on the stage, and you know that one of them has $1,000,000 and the other has $0.01 and they offer you the chance to trade cases, statistically, should you change cases or not? I content that at that point you have a 50/50 chance of your case containing $1,000,000, while my friend contends that there's a 1/50 chance of that, while if you were to trade cases, you would then have a 50/50 chance. Who is wrong? Your friend. Nah. You're wrong. Your friend is right. This is kind of still wrong, but shows it fine: http://www.youtube.com/watch?v=Zr_xWfThjJ0 No comprendo. Explain please. 1/2 > 1/26 Yeah, unless you change the parameters, I gotta go with this 1/26 is still applicable. 


At the time you choose between TWO cases, you have a 50/50 chance. Before the first, it was 1/50.



Originally Posted By HRomberg:
Originally Posted By CaptPlanet:
Originally Posted By Him:
Originally Posted By XterraJohn:
On the game show Deal or No Deal, where there's 26 cases that each contain a certain dollar amount and only one of them contains $1,000,000: Once you've opened all of the cases and you're down to two cases left, yours and the one on the stage, and you know that one of them has $1,000,000 and the other has $0.01 and they offer you the chance to trade cases, statistically, should you change cases or not? I content that at that point you have a 50/50 chance of your case containing $1,000,000, while my friend contends that there's a 1/50 chance of that, while if you were to trade cases, you would then have a 50/50 chance. Who is wrong? Your friend. Nah. You're wrong. Your friend is right. This is kind of still wrong, but shows it fine: http://www.youtube.com/watch?v=Zr_xWfThjJ0 No comprendo. Explain please. it makes no sense intuitively. i've seen the acutal mathematical proof though and its basically as presented in that video. if i knew what it was called, i'd google the proof. but i cant remember the actual name for the proof. to start you had a one in 3. you can improve your chances by picking the remaining choice you didnt take. if you do the test repetitively, you will always come out ahead. 


Originally Posted By Averagebear:
At the time you choose between TWO cases, you have a 50/50 chance. Before the first, it was 1/50. In statistics, the original population governs unless you change the parameters. I am having a hard time following when we are changing or not changing the parameters. Someone please help me with this. I'm confused on if the parameters are changed or not? If the original parameters are not changed than the results do not change. 


Originally Posted By CaptPlanet:
Originally Posted By HRomberg:
Originally Posted By CaptPlanet:
Originally Posted By Him:
Originally Posted By XterraJohn:
On the game show Deal or No Deal, where there's 26 cases that each contain a certain dollar amount and only one of them contains $1,000,000: Once you've opened all of the cases and you're down to two cases left, yours and the one on the stage, and you know that one of them has $1,000,000 and the other has $0.01 and they offer you the chance to trade cases, statistically, should you change cases or not? I content that at that point you have a 50/50 chance of your case containing $1,000,000, while my friend contends that there's a 1/50 chance of that, while if you were to trade cases, you would then have a 50/50 chance. Who is wrong? Your friend. Nah. You're wrong. Your friend is right. This is kind of still wrong, but shows it fine: http://www.youtube.com/watch?v=Zr_xWfThjJ0 No comprendo. Explain please. http://www.youtube.com/watch?v=mhlc7peGlGg Thanks! Learn something new every day and all that. 

Tree hugging Gun totin' Free love Capitalist Hippie
Yeah, that's how I roll. Do you hear the people sing.... 
Originally Posted By RDak:
Originally Posted By Averagebear:
At the time you choose between TWO cases, you have a 50/50 chance. Before the first, it was 1/50. In statistics, the original population governs unless you change the parameters. I am having a hard time following when we are changing or not changing the parameters. Someone please help me with this. I'm confused on if the parameters are changed or not? If the original parameters are not changed than the results do not change. The parameters changed because they eliminated 24 wrong choices. The intution is easier if you use a bigger number, say 1,000. 


Originally Posted By RDak:
Originally Posted By HRomberg:
Originally Posted By Him:
Originally Posted By XterraJohn:
On the game show Deal or No Deal, where there's 26 cases that each contain a certain dollar amount and only one of them contains $1,000,000: Once you've opened all of the cases and you're down to two cases left, yours and the one on the stage, and you know that one of them has $1,000,000 and the other has $0.01 and they offer you the chance to trade cases, statistically, should you change cases or not? I content that at that point you have a 50/50 chance of your case containing $1,000,000, while my friend contends that there's a 1/50 chance of that, while if you were to trade cases, you would then have a 50/50 chance. Who is wrong? Your friend. Two cases, P = 0.5, the other 24 are irrelevant. This. Right, if you traded cases you are at 5050. I am assuming you traded places. The parameters stay the same unless you change the original parameters. It seems to me that you have a 50/50 chance whether you trade cases or not. This is where the contention comes in. 


Originally Posted By XterraJohn:
Originally Posted By RDak:
Originally Posted By HRomberg:
Originally Posted By Him:
Originally Posted By XterraJohn:
On the game show Deal or No Deal, where there's 26 cases that each contain a certain dollar amount and only one of them contains $1,000,000: Once you've opened all of the cases and you're down to two cases left, yours and the one on the stage, and you know that one of them has $1,000,000 and the other has $0.01 and they offer you the chance to trade cases, statistically, should you change cases or not? I content that at that point you have a 50/50 chance of your case containing $1,000,000, while my friend contends that there's a 1/50 chance of that, while if you were to trade cases, you would then have a 50/50 chance. Who is wrong? Your friend. Two cases, P = 0.5, the other 24 are irrelevant. This. Right, if you traded cases you are at 5050. I am assuming you traded places. The parameters stay the same unless you change the original parameters. It seems to me that you have a 50/50 chance whether you trade cases or not. This is where the contention comes in. Watch the second vid. It's a better explanation. 

Tree hugging Gun totin' Free love Capitalist Hippie
Yeah, that's how I roll. Do you hear the people sing.... 
Originally Posted By Flats:
Originally Posted By RDak:
Originally Posted By Averagebear:
At the time you choose between TWO cases, you have a 50/50 chance. Before the first, it was 1/50. In statistics, the original population governs unless you change the parameters. I am having a hard time following when we are changing or not changing the parameters. Someone please help me with this. I'm confused on if the parameters are changed or not? If the original parameters are not changed than the results do not change. The parameters changed because they eliminated 24 wrong choices. The intution is easier if you use a bigger number, say 1,000. Thank you!! 5050. 


seems like if you are down to 2 its 50/50


Limit government not liberties

Your friend. Two cases, P = 0.5, the other 24 are irrelevant. This is the correct answer. It is known that the other 24 cases are empty. The remaining problem is the probability that the money is in one of the two remaining cases. This is coin toss. Don't over think the problem. :) 


In the situation the OP set up, there are only two cases left.. and we know from process of elimination that one is $1M and one is $0.01. You are basically asked to pick one.
Someone please explain this: How does the history leading up to this even matter? How is it NOT like a simple coin toss at that point? (seems like it would be 50/50) How is there EVER a 1/50 chance like some are saying? There are only 26 cases to begin with. how could your chances ever be worse than 1/26? ETA: first two questions answered 


this is not the same as the monte hall problem.
in the monte hall problem, you have a guy who knows where the goat is opening doors that don't have the goat. he's bringing information into the new setup. that's not happening here. 

Pick up that can.

Originally Posted By HRomberg:
Originally Posted By XterraJohn:
It seems to me that you have a 50/50 chance whether you trade cases or not. This is where the contention comes in. Watch the second vid. It's a better explanation. In the second video, the host knew which door did not contain the goat when he opened one. In the Deal or No Deal show, the person eliminating cases does not know which one is the winning case. 


Originally Posted By XterraJohn:
It seems to me that you have a 50/50 chance whether you trade cases or not. This is where the contention comes in. Yup, if you KNOW the other choices were "no reward" than it is 5050. 


Originally Posted By GARST:
In the situation the OP set up, there are only two cases left.. and we know from process of elimination that one is $1M and one is $0.01. You are basically asked to pick one. Someone please explain this: How does the history leading up to this even matter? How is it NOT like a simple coin toss at that point? (seems like it would be 50/50) How is there EVER a 1/50 chance like some are saying? There are only 26 cases to begin with. how could your chances ever be worse than 1/26? ETA: first two questions answered I messed up the question, friend and I were basing the argument off of 50 cases, but the actual game uses 26. Thanks for pointing that out, now off to edit the OP... 


Originally Posted By XterraJohn:
Originally Posted By HRomberg:
Originally Posted By XterraJohn:
It seems to me that you have a 50/50 chance whether you trade cases or not. This is where the contention comes in. Watch the second vid. It's a better explanation. In the second video, the host knew which door did not contain the goat when he opened one. In the Deal or No Deal show, the person eliminating cases does not know which one is the winning case. If the contestant chose all the cases to be opened, then you are correct. 

"During the second 100 days, we will design, build and open a library dedicated to my first 100 days." Barack Obama, May 9 2009

Originally Posted By XterraJohn:
Originally Posted By HRomberg:
Originally Posted By XterraJohn:
It seems to me that you have a 50/50 chance whether you trade cases or not. This is where the contention comes in. Watch the second vid. It's a better explanation. In the second video, the host knew which door did not contain the goat when he opened one. In the Deal or No Deal show, the person eliminating cases does not know which one is the winning case. If the person eliminating cases doesn't know then that changes the numbers (I have no idea what Deal or No Deal is). Edit: the odds are even if the eliminator has no knowledge. 


Originally Posted By Flats:
If the person eliminating cases doesn't know then that changes the numbers (I have no idea what Deal or No Deal is). Statistically, you should still trade, but your odds don't increase much. uhhh ... no 

Pick up that can.

Friend just called after having written a program to model the game and running it through several hundred thousand iterations; he conceded defeat and said that there was no significant statistical difference between trading cases or not.
Thanks for the help, everyone! 


Originally Posted By dogmeat:
Originally Posted By Flats:
If the person eliminating cases doesn't know then that changes the numbers (I have no idea what Deal or No Deal is). Statistically, you should still trade, but your odds don't increase much. uhhh ... no uhhh...yes. Whether the person is eliminating bad choices purposefully or not changes the odds. 


Originally Posted By Flats:
Originally Posted By dogmeat:
Originally Posted By Flats:
If the person eliminating cases doesn't know then that changes the numbers (I have no idea what Deal or No Deal is). Statistically, you should still trade, but your odds don't increase much. uhhh ... no uhhh...yes. Whether the person is eliminating bad choices purposefully or not changes the odds. prove it. 

Pick up that can.

With the Monty Hall example, the host knows which door the car is behind.
Are you still at an advantage to switch if he randomly chooses a door, and chooses a goat? 


Originally Posted By XterraJohn:
Friend just called after having written a program to model the game and running it through several hundred thousand iterations; he conceded defeat and said that there was no significant statistical difference between trading cases or not. Thanks for the help, everyone! Have him run the classic version as well. 

"During the second 100 days, we will design, build and open a library dedicated to my first 100 days." Barack Obama, May 9 2009

Originally Posted By AR4U:
Originally Posted By XterraJohn:
Friend just called after having written a program to model the game and running it through several hundred thousand iterations; he conceded defeat and said that there was no significant statistical difference between trading cases or not. Thanks for the help, everyone! Have him run the classic version as well. What is the classic version? 


If I'm incorrectly applying the Monty Hall thing, I apologize.



Originally Posted By xylo:
With the Monty Hall example, the host knows which door the car is behind. Are you still at an advantage to switch if he randomly chooses a door, and chooses a goat? in that case it doesn't matter. it's 50/50. 

Pick up that can.

Originally Posted By XterraJohn:
Originally Posted By AR4U:
Originally Posted By XterraJohn:
Friend just called after having written a program to model the game and running it through several hundred thousand iterations; he conceded defeat and said that there was no significant statistical difference between trading cases or not. Thanks for the help, everyone! Have him run the classic version as well. What is the classic version? The one described in the posted video, where the host is picking the doors. 

"During the second 100 days, we will design, build and open a library dedicated to my first 100 days." Barack Obama, May 9 2009

The 24 previous choices are irrelevant at that point. It is 50/50. Why is this difficult?


I live vicariously through myself.
In a past life, I was myself. 
Originally Posted By dogmeat:
Originally Posted By Flats:
Originally Posted By dogmeat:
Originally Posted By Flats:
If the person eliminating cases doesn't know then that changes the numbers (I have no idea what Deal or No Deal is). Statistically, you should still trade, but your odds don't increase much. uhhh ... no uhhh...yes. Whether the person is eliminating bad choices purposefully or not changes the odds. prove it. I have no idea what this show is, so here's the assumption I'm working off of. Contestant picks 1 case out of 26. Game show host opens 24 cases, one by one, until only 2 cases remain. Game show host knows which case contains the money. Given those conditions,there are 26 possible outcomes. 25 of them result in you getting no money. 1 results in you winning. The host is going to eliminate 23 bad choices no matter what you choose, and that's what increases your odds. Because he knows, there are 26 different ways to get to where you're down to 2 cases. Only one of them involves your initial choice being the correct one. So the odds are that you don't have the correct case, and you should switch. 


Originally Posted By Flats:
I have no idea what this show is, so here's the assumption I'm working off of. Contestant picks 1 case out of 26. Game show host opens 24 cases, one by one, until only 2 cases remain. Game show host knows which case contains the money. Given those conditions,there are 26 possible outcomes. 25 of them result in you getting no money. 1 results in you winning. The host is going to eliminate 23 bad choices no matter what you choose, and that's what increases your odds. Because he knows, there are 26 different ways to get to where you're down to your case and the winner. Only one of them involves your initial choice being the correct one. So the odds are that you don't have the correct case, and you should switch. The contestant opens the cases, not the host. 


Originally Posted By XterraJohn:
Originally Posted By Flats:
I have no idea what this show is, so here's the assumption I'm working off of. Contestant picks 1 case out of 26. Game show host opens 24 cases, one by one, until only 2 cases remain. Game show host knows which case contains the money. Given those conditions,there are 26 possible outcomes. 25 of them result in you getting no money. 1 results in you winning. The host is going to eliminate 23 bad choices no matter what you choose, and that's what increases your odds. Because he knows, there are 26 different ways to get to where you're down to your case and the winner. Only one of them involves your initial choice being the correct one. So the odds are that you don't have the correct case, and you should switch. The contestant opens the cases, not the host. Then like I said several posts up, the odds are 5050. 


Originally Posted By Flats:
Originally Posted By dogmeat:
Originally Posted By Flats:
Originally Posted By dogmeat:
Originally Posted By Flats:
If the person eliminating cases doesn't know then that changes the numbers (I have no idea what Deal or No Deal is). Statistically, you should still trade, but your odds don't increase much. uhhh ... no uhhh...yes. Whether the person is eliminating bad choices purposefully or not changes the odds. prove it. I have no idea what this show is, so here's the assumption I'm working off of. Contestant picks 1 case out of 26. Game show host opens 24 cases, one by one, until only 2 cases remain. Game show host knows which case contains the money. Given those conditions,there are 26 possible outcomes. 25 of them result in you getting no money. 1 results in you winning. The host is going to eliminate 23 bad choices no matter what you choose, and that's what increases your odds. Because he knows, there are 26 different ways to get to where you're down to 2 cases. Only one of them involves your initial choice being the correct one. So the odds are that you don't have the correct case, and you should switch. My uhhh no was in reference to this: "Statistically, you should still trade, but your odds don't increase much." if the person opening doesn't know where the prize is (e.g. it's the contestant) then at the end it's 50/50. 

Pick up that can.

Originally Posted By dogmeat:
Originally Posted By xylo:
With the Monty Hall example, the host knows which door the car is behind. Are you still at an advantage to switch if he randomly chooses a door, and chooses a goat? I think so. Yes, because your odds at that point are 2/3 compared to the original 1/3 odds of the first door you chose. In the the OP's example, the odds for the case you originally chose is 1/26, the likelihood of picking the winning case doesn't change as you open cases (As long as it's you opening the cases and not the host). Assume 'N' number of equal possibilities, each case opened is 'N1', all the possibilities remain equally as likely. 


I am not sure where the 26 cases come into play. You know there are two cases that contain $X. Therefore there is a 5050 chance on getting the higher dollar amount.



Originally Posted By PattyOSullivan:
I am not sure where the 26 cases come into play. You know there are two cases that contain $X. Therefore there is a 5050 chance on getting the higher dollar amount. Incorrect. It's 1/26, the likelihood of having the winning case never changed. 


Originally Posted By KellenerSptM5:
Originally Posted By dogmeat:
Originally Posted By xylo:
With the Monty Hall example, the host knows which door the car is behind. Are you still at an advantage to switch if he randomly chooses a door, and chooses a goat? I think so. Yes, because your odds at that point are 2/3 compared to the original 1/3 odds of the first door you chose. In the the OP's example, the odds for the case you originally chose is 1/26, the likelihood of picking the winning case doesn't change as you open cases (As long as it's you opening the cases and not the host). Assume 'N' number of equal possibilities, each case opened is 'N1', all the possibilities remain equally as likely. I edited that out. The "I think so" was a mistake. Whether or not monte knows what is behind the door he opens before opening it matters. if he knows and opens a goat, you should switch, if he gets the goat randomly, it's like the OPs scenario and you're at 50/50. 

Pick up that can.

Originally Posted By dogmeat:
My uhhh no was in reference to this: "Statistically, you should still trade, but your odds don't increase much." if the person opening doesn't know where the prize is (e.g. it's the contestant) then at the end it's 50/50. Sorry, I edited that as soon as I posted. Just not fast enough. 


Some of you guys are way over thinking this. It doesn't matter if there were 10, 26, 50, or 5 million cases to start with, there are only TWO left. Get it, two out of the original 10 million, it doesn't matter. So you only have TWO choices at this point, and you know what each one could be. Flip a coin, same odds.



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