Drawing straws, is it fair?

The idea of drawing straws is used as a "fair" way to choose one person from a group to do an unsavory activity. I recently saw this in a movie and was wondering if this was actually fair.

With n people there are n straws, n-1 normal straws and 1 short straw. The person who picks the short straw loses, and there is no replacement of straws. So the first person trying to pick any of the n-1 long straws from the n straws, person 2 picks n-2 from the n-1 straws and so on. Of course the next person only goes if the person before them succeeds, picks a long straw.

First off is this fair, aka does the order you pick in not matter, secondly if it is not fair is there a formula for the probability of each spot (I have a feeling it is something like the probability of success of all previous persons x prob of drawing a long straw) ?

Re: Drawing straws, is it fair?

Yes, the process that you mentioned would be unfair, since n > n-1 (the number of choices) -> 1/n < 1/n-1 (the probability). also, since there's only one short stick, if it's chosen early, the other participants who haven't chosen don't risk anything; they win automatically

A more fair process would be for everyone to draw a stick simultaneously from the group. [then again, there might be secrets markings on the tops of these sticks that certain people know about so they never choose the short one ;) ]

Re: Drawing straws, is it fair?

doesn't this overlook the fact that person 1 has a 100 percent chance of picking a straw and everyone else has a lesser chance?

Re: Drawing straws, is it fair?

I don't understand what you mean. The first person to choose is in favor, since there are more possibilities than second, third, etc.

Re: Drawing straws, is it fair?

i understand the fact that n-1/n > n-2/n-1 . But don't you have to account for the fact that there is a 1/n chance the game ends after the first person, person 1 has a 1/n chance of failure, and person two has an n-1/n chance of going and a 1/n-1 chance of failure.

Re: Drawing straws, is it fair?

It is not necessarily a sequential game. Why would it be different if everyone picked a straw simultaneously? If all closed their eyes and picked a staw, revealing all at the same time, it is the same. Doing it sequentially only saves time if the short straw goes early.

Re: Drawing straws, is it fair?

So I worked through a solution and from what I can tell it yields a fair result. Let me know what you think.

P(drawing the short straw) = P(short straw from given straws) * P(attempting)

The probability of attempting is equivalent to the person before you drawing a long straw, the game continuing to your draw

thus

P(drawing the short straw) = P(short straw from given straws) * P(previous person drawing a long straw)

writing P(X=1,2,3...n) indexing the probability of drawing the short straw from each respective position

given b=1,2,3...n ; a=1,2,3...n-1

P(X=$\displaystyle X_1$) = (1/n)*1

P(X=$\displaystyle X_b$) = (1-$\displaystyle X_ {b-1}$) * 1/(n-a)

For example n=3

$\displaystyle X_1$=(1/3)*1= 1/3

$\displaystyle X_2$=(1-(1/3))*(1/2)=1/3

$\displaystyle X_3$=(1-(2/3))*1 = 1/3

Re: Drawing straws, is it fair?

Assuming nothing shady is going on, the idea behind drawing straws is obviously fair. You might say person 1 has an advantage because the probability of them picking the short straw is 1/n whereas it is 1/(n - 1) for person 2, but this is wrong because it is (incorrectly) conditioning on the fact that person 1 has not drawn the short straw.

I think TKHunny's post makes this clear enough, so I only post because someone else has posted something that is incorrect :) Whether you do this sequentially or not should clearly make no difference, and the simultaneous game is clearly fair.