The con is on

[pozorvlak]

One of the things my new flatmate (whodo_voodoo, for those of you keeping track) brought with him is a television. I'm a bit dubious, as my ability to work is low enough as it is, but it did mean that I was finally able to catch an episode of Hustle, the "complicated confidence trick" series starring Robert Vaughn. I love crime capers even more than I love kung fu movies, so it was deeply frustrating to me that there was a new one going out on telly every week and I was missing it. Anyway, I struck particularly lucky with this episode: halfway through, an actual guy with a katana came in and got all kenjutsu on the gang's asses. Result, quite frankly.

The katana dude was seeking revenge for his father's death (by seppuku, natch), and so he forced the gang to eat pieces of possibly-contaminated fugu one by one. There was much debate about who was the leader (I'm Spartacus!), and so should go first, and where one should go in the order to maximise one's chances of survival. As much debate as you can have while a scary Japanese guy is threatening you with a sword, that is. The general consensus was that it was best to go early, while you still had a decent number of fugu pieces to choose from.

[The mathematically-inclined may like to look away now. Sorry about lack of cut, but I needed one for the spoilers, and you can't nest them AFAIK.]

Here's the thing: if the order is chosen in advance, then it doesn't matter where you go. Suppose the gang has m members, and there's one piece of fugu each. Only one is poisonous. If you go first, you have a one in m chance of choosing the poisoned piece. If you go second, you have a one in (m-1) chance of choosing the poisoned piece when it's your go, but you'll only get to your go if the first guy survives. So your chance of dying is (m-1)/m * 1/(m-1) = 1/m, the same as the first guy. If you go in the n'th place, you'll only die if you choose the bad piece and everyone in front of you chose good pieces, so your chance of dying is (m-1)/m * (m-2)/(m-1) * ... * (m-n+1)/(m-n+2) * 1/(m-n+1) = 1/m.

If the order isn't fixed, however, it gets a bit more confusing. When there are n pieces left, you've got a 1/n chance of picking the bad one, by the reasoning above: obviously, this is going to go up as n goes down. But at each stage, choosing to eat now or wait doesn't affect your odds of survival. Maybe you can't choose not to die, but only choose the manner of your dying: an appropriately Bushidoesque thought.

Whoever said probability was easy?

Comments

[mi-guida.livejournal.com]

Yay, unto Hustle! I may have tot try to find it elsewhere.... I loved watching it.

[bronxelf-ag001.livejournal.com]

I am SO calling you when I have to go back to school and take all those arch. calc. classes.

[pozorvlak.livejournal.com]

Sure thing :-)

Complications

[neoanjou.livejournal.com]

How would it be affected if one were able to (to a certain degree) alter the chances in your favour - i.e. if I know a little something about fugu, for isntance that the darker the flesh colour the more likely poision is found there, meaning that for every 'n' the chance of you picking the poisioned fish is slightly less than 1/n and the chance of picking the others is therfore slightly greater than (1-1/n).

I suspect that would make it better to go first, but I haven't looked at the matematics of it.

Re: Complications

[pozorvlak.livejournal.com]

Do the others also have this knowledge?

Re: Complications

[neoanjou.livejournal.com]

Try both ;) ...

Maybe I have the solution for everyone having it though -

In the case of perfect knowledge on the part of the participants - for instance mathematicians picking a number between 1 and 10 and the one who gets the perfect number is killed - then obviously its best to go within the first nine.

In the other limiting case of no knowledge (as above) it doesn't matter where you go.

I'd guess that with knowledge between these two limits its still better to be early in the choosing - ideally first, but the advantage is not as good as the perfect knowledge limit.

In the case of only one person having any knowledge I suspect the same will apply - if his knowledge is perfect then he'll want a choice - i.e. to be anywhere in the first n-1 people - if his knowledge is not perfect then he'll still want to be early to have as great a choice as possible.

[totherme.livejournal.com]

What if the poison were slow acting, so everyone would have eaten by the time the first guy died?

[pozorvlak.livejournal.com]

Shouldn't make a difference, if there's only one poisoned piece - once it's eaten, it's removed from play. If there are several pieces, then that does change the game - several people can die rather than just one.