You are shown four, identical envelopes on a table. You are told that one contains a $100 bill, the rest are empty. You are told to choose one and pick it up. Then, two of the three envelopes still on the table are removed and you are told that they were empty. Now, do you keep the one in your hand, exchange it for the one still on the table or does it not make any difference?
Hold the one in my hand up too the light and then decide 
By rules of statistics, you should ALWAYS trade envelopes.
The envelope in your hand has a 1 in four chance of winning, while the envelope on the table has a 1 in 2 chance of winning.
Maroon92 wrote: By rules of statistics, you should ALWAYS trade envelopes. The envelope in your hand has a 1 in four chance of winning, while the envelope on the table has a 1 in 2 chance of winning.
Except that when the two envelopes were removed and you were told that there was no money, the chances that your envelope was right immediately changed from 25% to 50%.
You have a 1 in 2 chance whether you exchange or not.
Where the odds get interesting is if the envelopes all have different values in them- which is a game show called "lets make a deal". Then the odds are very fluid based on the known values or not. Those odds are also calculated with the value on the table.
Maroon92 wrote: By rules of statistics, you should ALWAYS trade envelopes. The envelope in your hand has a 1 in four chance of winning, while the envelope on the table has a 1 in 2 chance of winning.
I am pretty sure you're getting some stuff mixed up there...
No, he's right - according to statistical reasoning. But by the rules of life, it makes no difference. You're never going to win the $100. You were not the 15th caller. No Nigerian official really wants you to stash his millions.
nderwater wrote: No, he's right - according to statistical reasoning. But by the rules of life, it makes no difference. You're never going to win the $100. You were not the 15th caller. No Nigerian official really wants you to stash his millions.
Well, I don't see how, unless by "he" you meant "Alfa".
Based on years of experience, my odds of winning free money are zero, so I know that whichever I keep will be empty.
I am not assuming that I get to keep the $100, even if it is in my envelope. 
Maroon92 wrote: By rules of statistics, you should ALWAYS trade envelopes. The envelope in your hand has a 1 in four chance of winning, while the envelope on the table has a 1 in 2 chance of winning.
Right answer, numbers a little off.
When you pick up one envelope, 25% probability is in your hand, 75% is still on the table. Removing the two empties doesn't change that. You already knew at least two were empty. So you swap.
Can I trade the envelope for what Jay has in the box Monte?
tuna55 wrote:nderwater wrote: No, he's right - according to statistical reasoning. But by the rules of life, it makes no difference. You're never going to win the $100. You were not the 15th caller. No Nigerian official really wants you to stash his millions.Well, I don't see how, unless by "he" you meant "Alfa".
Would you rather pick up the beer glass that is ¼ full or ½ full?
alfadriver wrote:Maroon92 wrote: By rules of statistics, you should ALWAYS trade envelopes. The envelope in your hand has a 1 in four chance of winning, while the envelope on the table has a 3 in 4 chance of winning.Except that when the two envelopes were removed and you were told that there was no money, the chances that your envelope was right immediately changed from 25% to 50%. You have a 1 in 2 chance whether you exchange or not. Where the odds get interesting is if the envelopes all have different values in them- which is a game show called "lets make a deal". Then the odds are very fluid based on the known values or not. Those odds are also calculated with the value on the table.
Common sense says that you are right. When this test is repeated many times in real life, the percentages come out to 25% and 75%.
"Yes, it is counterintuitive, but think of it this way. The probability that you picked the correct one does not change just because two empty envelopes were revealed. The probability you picked the correct envelope is still 25%. The probability it is one of the other 3 is 75%. It just so happens you know it is not in 2 of the 3. Thus the probability is 75% it is in the remaining one."
Here's another head-sploding exercise: http://www.kiekeben.com/envelope.html
Of course, rules of statics are simply mathematical models.
Blowing this one out a little... Let's say that there were a million envelopes to start with and you took away 999,998 and showed they were empty. By the statistical model, if you switched envelopes you'd have a 99.9998% chance of getting the $100.
Yeah, right.
I agree with intuition, that 'past occurences do not affect current probabilities'. At the end of the day you're stuck with two choices - 50/50.
I'm no mathmatician but I'll go out on a limb and say that math DOES always make sense and that intuition has nothing to do with it.
chuckles wrote: I'm no mathmatician but I'll go out on a limb and say that math DOES always make sense...
I'm almost done with my math major, and math does NOT always make sense.
alfadriver wrote:Maroon92 wrote: By rules of statistics, you should ALWAYS trade envelopes. The envelope in your hand has a 1 in four chance of winning, while the envelope on the table has a 1 in 2 chance of winning.Except that when the two envelopes were removed and you were told that there was no money, the chances that your envelope was right immediately changed from 25% to 50%. You have a 1 in 2 chance whether you exchange or not. Where the odds get interesting is if the envelopes all have different values in them- which is a game show called "lets make a deal". Then the odds are very fluid based on the known values or not. Those odds are also calculated with the value on the table.
Go watch 21! NOW!
mtn wrote:alfadriver wrote:Common sense says that you are right. When this test is repeated many times in real life, the percentages come out to 25% and 75%.Maroon92 wrote: By rules of statistics, you should ALWAYS trade envelopes. The envelope in your hand has a 1 in four chance of winning, while the envelope on the table has a 3 in 4 chance of winning.Except that when the two envelopes were removed and you were told that there was no money, the chances that your envelope was right immediately changed from 25% to 50%. You have a 1 in 2 chance whether you exchange or not. Where the odds get interesting is if the envelopes all have different values in them- which is a game show called "lets make a deal". Then the odds are very fluid based on the known values or not. Those odds are also calculated with the value on the table.
The problem is that once the two envelopes are removed, not having money, the odds change for all envelopes. Those two removed become 0/4, therefore both of the remaining two are 1/2.
It's true that the odds change for the remaining envelope, but one has to remember that the odds change for your's too.
The other game show is called "deal or no deal". It's all about changing values to choices.
nderwater wrote: Blowing this one out a little... Let's say that there were a million envelopes to start with and you took away 999,998 and showed they were empty. By the statistical model, if you switched envelopes you'd have a 99.9998% chance of getting the $100. Yeah, right.
That actually is exactly right.
The odds would be the same only if you randomly selected two more envelopes... but you don't - you specifically throw-out ONLY non-winning options.
Think of delicious delicious cake with a cherry somewhere on top. Cut the cake in unequal halves, and the cherry is more likely to be on the bigger half. Now cut the bigger half down to the same size as the smaller peice, but only cut away parts that definitely don't have they cherry on top. Even though you have two equal-sized peices, because the other peice WAS bigger, it has an "unfair advantage" in terms of odds of having the cherry.
Same with the envelopes... even though you threw out 999,998 envelopes, because you KNEW that you were keeping the winner, the envelope on the table doesn't represent one envelope, it represents ALL of the envelopes that were in it's "half" in the original split.
These are statistics bases on gambling. They aren't supposed to make sense.
EastCoastMojo wrote: I am not assuming that I get to keep the $100, even if it is in my envelope.
You are correct. After taxes you'd get to keep about $12.95
N Sperlo wrote:tuna55 wrote:Would you rather pick up the beer glass that is ¼ full or ½ full?nderwater wrote: No, he's right - according to statistical reasoning. But by the rules of life, it makes no difference. You're never going to win the $100. You were not the 15th caller. No Nigerian official really wants you to stash his millions.Well, I don't see how, unless by "he" you meant "Alfa".
Doesn't matter. In either case, it's clearly time for more beer.
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