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### What if scenarios

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Dobagoh

Member

Quote:
Mar The Dragon:
I know that. That means that that is one of the 2 boxes full of rocks, so either the one I have or the second one he has if full of gold and the other has rocks in it. So therefore, between the two unopened boxes, there is a 50/50% chance of the one that I chose to have gold in it.

No, there is only a 1/3rd chance the box you're holding has the gold in it.

--------
Play the game in real time.
Possible mix ups of boxes:
A- 0 | A- 1 | A- 0
B- 0 | B- 0 | B- 1
C- 1 | C- 0 | C- 0

In column 1, you picked A.
B is eliminated. You switch, you win; don't switch, you lose
In column 1, you picked B.
A is eliminated. You switch, you win; don't switch, you lose
In column 1, you picked C.
One of A or B is eliminated. You switch, you lose; don't switch, you win
In column 2, you picked A.
One of B or C is eliminated,. You switch, you lose.; don't switch, you win
In column 2, you picked B.
C is eliminated, you switch you win; don't switch, you lose
In column 2, you picked B.
C is eliminated, you switch you win; don't switch, you lose
In column 3, you picked A.
C is eliminated, you switch you win; don't switch, you lose
In column 3, you picked B.
A or C is eliminated, you switch, you lose; don't switch, you win
In column 3, you picked C.
A is eliminated, you switch you win; don't switch, you lose

By switching, you win 6 times and lose 3 times.
By not switching, you win 3 times and lose 6 times.

It is clearly better to switch than to not switch.

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DirtyEarsBill

Senior Member

To help you better grasp the logic behind switching, imagine if there were a million boxes instead of 3. Upon choosing one, 999,998 of them are opened up, all containing rocks. Do you stick with your original choice, or do you switch to the other unopened box?

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Ryuna

Recruiter

Or the guy with the boxes were lying and all three were rocks.

Win chance - 100% because you still won rocks.

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Mar The Dragon

Senior Member

Quote:
Dobagoh:
Here let me spell it out for you.

You pick box 1. Guy eliminates a random box.

If box 1 has gold, you switch to a box with rocks, you lose.

If box 1 has rocks, you switch to 3rd box, you win.

If box 1 has rocks, you switch to 3rd box, you win again. (2 possible rock boxes to eliminate, so you have two outcomes)

Switching gives you 2/3 chance of winning, not switching gives you 1/3 chance of winning.

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Other way:

You random pick a box, 1/3 chance of winning
One box is eliminated.
You now have 1/2 chance of winning by switching.

Your chances 'improved' by 1/6, why would you not switch?

1/2 chance of having the gold box. Switch or no switch, I have the same chance of getting the gold box, so my chances would not improve. I would still have a 50% chance of getting the box with gold in it.

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DirtyEarsBill

Senior Member

Brawndo has what plants crave.

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Dobagoh

Member

Quote:
Mar The Dragon:
1/2 chance of having the gold box. Switch or no switch, I have the same chance of getting the gold box, so my chances would not improve. I would still have a 50% chance of getting the box with gold in it.

You don't have 1/2 chance of having the gold box. Please review the chart.

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MewOfDoom

Senior Member

Mar is correct in this case because of a technicality.

The only way the percentage would go up would be if the person opening the rock box *knew* which boxes contained what, and this is not stated in the problem. As long as the other person in the equation also is unaware which box contains the gold, it doesn't help one way or the other to switch.

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Mar The Dragon

Senior Member

Quote:
Dobagoh:
You don't have 1/2 chance of having the gold box. Please review the chart.

Not at first you don't, but after the box is eliminated, it is as if there were only 2 boxes to begin with. You could have chosen the gold box, or you could have chosen the rock box. Since there are only two options left, it is a 50/50% chance, and swapping would not increase those chances beyond 50/50%. The only point I aw trying to make is that after the box is eliminated, it is a 50/50% chance.

A- 0 | A- 1 | A- 0
B- 0 | B- 0 | B- 1
C- 1 | C- 0 | C- 0

Since one of the two rock boxes is always eliminated, it is like there was no third box, so take out column 1 and row C.

A- 1 | A- 0
B- 0 | B- 1

You have a 50/50% chance of having the correct box, switch or no switch.

EDIT: Thank you, MewOfDoom, that is the point I was trying to make.

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DirtyEarsBill

Senior Member

I guess you guys really like rocks.

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Mar The Dragon

Senior Member

Although... he never said what kind of rocks they were. Considering that there is a possbiltity of the rocks being moon rocks, Mars rocks, or any other type of rock that is extremely rare or valuable, the box of rocks would be far more valuable than the box of gold.