# Drawing straws, is it fair?

• Jul 2nd 2011, 04:55 PM
epreble
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) ?
• Jul 2nd 2011, 05:03 PM
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 ;) ]
• Jul 2nd 2011, 05:05 PM
epreble
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?
• Jul 2nd 2011, 05:08 PM
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.
• Jul 2nd 2011, 05:24 PM
epreble
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.
• Jul 2nd 2011, 06:35 PM
TKHunny
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.
• Jul 2nd 2011, 07:10 PM
epreble
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
• Jul 3rd 2011, 07:52 AM
Guy
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.