So, one hundred Gnomes are going on vacation....

SlowToady · 08:50 AM

and they have their passports and things all ready, when all of a sudden, they are kidnapped by Evil Trolls. The Gnomes are (conveniently) named 1 through 100. Now, the Evil Trolls take the passports of the Gnomes, and put them into a cave. The Evil Trolls then explain the rules to the following game they are going to play with the Gnomes. Each Gnomes passport is put into a separate box, labeled 1 through 100. The Gnomes are brought in to the cave containing the boxes one at a time and allowed to open at most 50 boxes. If the Gnome sees his passport, this is good. If not, then the Evil Trolls will eat the Gnomes for dinner. If only 1 Gnome doesn't find his passport, all of them are eaten for dinner. The passports are all put back, and all the boxes closed prior to each Gnome entering. The Gnomes are not allowed to communicate in any way once the first Gnome has entered the cave.

What is the probability that the Gnomes are eaten for dinner?

onlycafe · 10-27-2008, 07:59 PM

50/50.

Noah930 · Noah930's Garage

Not very good. Gnomes taste awful.

slodave · slodave's Garage

Just as long as this little dude survives.

surflvr911sc · surflvr911sc's Garage

on2wheels52 · on2wheels52's Garage

Near 100%. I wonder how the trolls cook things like that.
Jim

sketchers356 · 10-28-2008, 04:35 AM

1-.5^100=1-7.9*10^-31

So they are screwed.

legion · 10-28-2008, 05:19 AM

I get that the gnomes have a 0.0000000000000000000000000000788860905% chance of being successful.

robbajob · 10-28-2008, 06:25 AM

Do the numbers on the passports correspond with the number of the box? You don't say that they're mixed up, so I'm going for 0%.

Jim Bremner · 10-28-2008, 07:20 AM

I want my click back.

Jim Richards · 10-28-2008, 07:32 AM

Gnomes taste like chicken.

LeRoux Strydom · 10-28-2008, 07:41 AM

The probability of 1 gnome finding his passport is 0.5 (50%). The probability of all 100 gnomes finding their passports is 0.5^100=7.89e-31. The probability of the gnomes being eaten is 1-0.5^100 = 0.9999999999999999999999999921, or as near to 100% as dammit is to swearing.

They're screwed.

legion · 10-28-2008, 07:43 AM

I still think the gnomes have better odds than winning the lottery...

SlowToady · 10-28-2008, 08:03 AM

sketchers and LeRoux, those probabilities are correct if the Gnomes pick the boxes at random. But, they can do better than that. Actually they can do quite a bit better, although I will say that it's still most likely that they get eaten.

Robbajob, there's a picture on the passports (the owners picture). The passports are placed in the boxes at random, and so no, there is no correspondence between the Gnomes and the numbers on the boxes. However, their chances for survival are still ultimately better than 0%.

Onlycafe is closest, but still incorrect.

Isn't this fun?!

robbajob · 10-28-2008, 08:07 AM

I was saying their chances of getting eaten were 0%.

I thought I'd point that out just in case the little fellas, by some miracle, all find their passports and prove me right.

SlowToady · 10-28-2008, 08:13 AM

Ah, sorry, I was reading quickly and mis-understood your post. Sorry! I've left my original comments in case in case it was originally ambiguous.

Quote:

Originally Posted by robbajob

I was saying their chances of getting eaten were 0%.

I thought I'd point that out just in case the little fellas, by some miracle, all find their passports and prove me right.

LeRoux Strydom · 10-28-2008, 08:39 AM

The gnomes can only improve their chances if they can cooperate by opening the boxes in a certain sequence. Since they cannot communicate, how will they determine the order in which to open the boxes?

SlowToady · 10-28-2008, 08:50 AM

Ooops sorry, the statement wasn't as clear as I thought. The Gnomes cannot communicate after the first Gnome has been brought to the cave.

So the Trolls explain the game and the rules, and the Gnomes can communicate, but once the Trolls take the first Gnome into the cave, no more communication.

Sorry

edit: I guess we could also say the Gnomes are super smart and all figured out the correct sequence on their own.

Quote:

Originally Posted by LeRoux Strydom

The gnomes can only improve their chances if they can cooperate by opening the boxes in a certain sequence. Since they cannot communicate, how will they determine the order in which to open the boxes?

gtc · gtc's Garage

Are you asking for the proability of all gnomes being eaten, or the probability of any one given gnome being eaten?
FWIW, the probability of a gnome finding his passport is better than 50%. I get 68.82%.

71T Targa · 10-28-2008, 09:49 AM

Quote:

Originally Posted by gtc

Are you asking for the proability of all gnomes being eaten, or the probability of any one given gnome being eaten?
FWIW, the probability of a gnome finding his passport is better than 50%. I get 68.82%.

If one is eaten, they all are.

How do you get 68.82%? There are 100 boxes, and you can open 50. Looks a lot like 50/100 to me.