Greetings,

I was wondering if the collective genius of STW could put my colleagues and I out of our misery: after much meandering debate and lack of knowledge we seem have to have reached the conclusion that drawing straws is actually an unfair way of selecting someone from a group. The first person that draws a straw has a higher chance of drawing a long straw than each successive person – this makes sense to us, possibly…

I then confused things by asking if the same process was followed, but in the pitch black, and the results only revealed when everyone had drawn – it seemed logical to me that if there were the same number of straws and people then it MUST be a fair way of deciding.

I haven't explained this very well and was having trouble explaining why, at the time, I was sure this method is fair. I think my problem is that drawing straws has been around for so long, and survived, that it just has to be fair and I can't accept that it can't be.

Could you wise folk set us straight please…?

Depends whether you are on a conveyor belt at the time surely?

Hi,

The first person that draws a straw has a higher chance of drawing a long straw than each successive person

FAIL 🙂

everyone knows paper-scissors-stone is the way to decide who does something.

failing that, thumbwar.

and yeah, you need a better grasp of probabilities.

there is a larger chance of getting the short straw each time but also less in the game – so it evens out? Also there are two outcomes to ewach pick so that alters things

so person 1 picks – its a 4 / 5 chance that they don't get the short straw so 20% chance they do

Person two picks – its a 1/ 5 chance the game is over before they pick and a 3/ 4 chance that they don't get a long straw
Person 3 picks – 2/5 chance the game is over before they join and 2/3 chance tehy get a long straw.

Hmmmmmmmmm

That don't make sense either something flawed in my calculations but there are two things to take into account – the odds of the gae being over before they join and the odds of them picking the short straw. I think you need to use teh odds of nothing happening as well and multiply them together.

I see what you mean in that if there are 6 people then the first person has a 1 in 6 chance of getting the short straw…so by the time you're on the fifth person there's a 1 in 2 chance of getting the short straw.
However at the same time the fifth person has less chance of actually having to pick a straw at all (assuming that you stop if the short straw is picked). If that makes any sense at all! So I think it is fair.

Thanks for the replies so far (apart from allthepies: "FAIL" with no explanation is no answer at all, in fact, I think it's actually a fail…)

The idea that people who go later in the game might not need to draw at all seems to make sense! Maybe…

EDIT: also, Jambo, I may indeed need a better grasp of probabilities, but if you could enlighten us all as to where our knowledge falls short, that would be gratefully received.

Every participant has the same chance of ending up with the short straw is actually my answer then 🙂

take time out of it, there are three straws, and each person gets a straw whether they choose it or are left with it. One of those straws is the bogey, so everyones got a one in three chance of getting it.

http://www.cut-the-knot.org/Probability/ConditionalProbability.shtml

Aye, the problem is that assumes simultaneous selection of the straws.
When the number of straws is reduced and the short one hasn't been selected, the probability of selecting it the next time increases.

just do…..ip dip

can't we get some quantom physics in here?

if everyone draws their straws and doesn't observe them, are they both long and short straws….

Rock paper scissors is only worth using if you employ the spock lizard expansion

Blast it, now I'm even more confused…

Allthepies (or anyone!): can you explain why the people who draw earlier DO NOT have an advantage…?

Jam bo: an excellent answer sir, well redeemed. (Although we do rather need to know the state for the game to work, but I guess, by not looking, no cats are harmed in the making of this decision. Or are they…?)

Ip dip and R/P/S are fine suggestions, however, I am rather interested in the validity of drawing straws.

Rock on, peeps, rock on.

Imagine the straws are all the same length, but inside one straw is a note saying "FAIL". You are only allowed to look inside your straw once all straws are drawn.

Now play the game again.
Now play it again this time you can look inside your straw as soon as you draw it……has anything changed?

You need to look at the possible outcomes for each person.
Person 1 has a 1 in 3 chance of pulling the short straw.

Person 2 the outcomes are
Person 1 already has the short straw, in which case they have a long straw.
person one has a long straw, they draw the remaining long straw
person 1 has a long straw, they draw the short straw
So a 1 in 3 chance of getting the short straw.

Person 3
Person 1 already has the short straw, in which case they have a long straw.
Person 2 already has the short straw, in which case they have a long straw.
They have the short straw
So a 1 in 3 chance of getting the short straw.

If there are 6 people playing the game you have a 1/6 chance of drawing the short straw regardless of your order in the draw.
For the first person it is a straight 1/6.
For the last person there a 5/6 chance you don't have to draw because someone else lost

Think of it this way: There are 6 straws and you are the last person to draw. There's a 5/6 chance that the short straw has already been drawn. Therefore you have a 1/6 chance of drawing the short straw as the last person. Which is the same as the first person. So its all fair…

j_me: spooky – we wrote the same thing at the same time, even using the same number of straws… what's the odds of that?

1/6 ?
🙂 that is quite spooky

can you explain why the people who draw earlier DO NOT have an advantage…?

Because they all 'entered the room' with an equal chance of drawing the short straw

Think russian roulette – each individual has a 1/6 chance of being shot – the only thing that resets that chance is spinning the chamber (or in your case, redrawing the straws)

If there are 6 people sitting at the table, and you spin the chamber every time and everyone gets a go, then you have a 1/6 chance of getting ventilated each time, this only changes if you agree that you will end the game as soon as someone is shot – in that scenario, going last means you have a greater chance of survival, as there is a chance that it will never get to your turn – but, then, thats not how you play the game,,,,

if you hand the pistol round without spinning the chamber, then there is a 1/6 chance that you, as an individual will get killed – although once you've got through five people unscathed, the outlook for the sixth person is really not looking good, the probability that he ever gets to pull the trigger is equal to the probability of the first five not being killed, so still 1/6.

so, the "chance" is reset by the spinning of the chamber, not by each attempt.

Say you have 5 people.

If we work out the probability that the 4th person draws a short straw then we say:

P(A) = First 3 people didn't draw a straw
P(B) = 4th person draws a straw

P(A) = (Probability of drawing a long straw) / (Probability of having a go)
P(A) = (0.8 * 0.6 * 0.4) / (1 * 0.8 * 0.6) = 0.4

So the probability that the 4th person draws the straw given that the previous 3 didn't is:

P(B|A) = 0.5

So the probability of A and B being true is:

P(A and B) = P(A) * P(B|A) = 0.4 * 0.5 = 0.2

Therefore the probability of drawing a short straw for person 4 is 20%, same as the first person. (Yes I did borrow and then modify and further explain this from elsewhere :p)

You're looking at it the wrong way.

With say six people, every person has a 1 in 6 chance of drawing the short straw. After you've started selecting, the straws drawn don't change those odds, even if if seems like it does.

If it helps, think about this; everyone draws a straw blind, without looking at it, then all compare their choices simultaneously at the end. Seem fair? Of course it is. The odds haven't changed just because you've got your eyes shut.

chvck

Nice one – thats what I was trying to get to but couldn't quite make it.

Thank you, gang, I think it's finally been explained to my satisfaction and in such a way that I can pass on the explanation to straw-drawing-doubters.

Once again, the collective might of STW's hivemind wins the day – cheers everyone! 🙂

The plot thickens, as does my skull….

The above explanations seem to ride upon the game ending as soon as the short straw is drawn – the argument raging here is now concentrating on the game continuing until all straws are drawn and THEN revealing who has what.

This would seem to remove the advantage of the people who draw later (where previously they had a diminishing chance of even having to draw).

Does this affect the odds? Or does the chance of a previous drawee having already removed the short straw affect the later drawees' probability of not drawing the short one? Is anyone still reading at this point!?

Drink him make can't you but, water to horse a lead can you.

bananaworld – Member

"This would seem to remove the advantage of the people who draw later (where previously they had a diminishing chance of even having to draw)."

No. Why would knowing the results change the results?

Rock Scissors Stone – I thought Rock was Dead.

Does this effect the odds?

What happens if you draw straws on a conveyorbelt?

Plug it back into the above equations but remember to include every possibility (i.e. there is always a chance of 1 of the second person having a go but the probability of drawing a straw is going to affect later values depending on the people before you so you need to include all scenarios), if you plug the values in correctly then you'll get the same result that the chances are all even. I cba to plug all those values in to prove it though.

Edit: I'm allowing for multiple drawings of short straws there, i.e. 2 people can be the unlucky suckers

Stop the game when short drawn or conceal and then all reveal at the end. Makes no difference to the individual odds.

It makes no difference whether you stop the game or not, or whether you reveal straws.

Once the short straw has been drawn, nobody else can draw it. This is the case whether or not the later people know that it has already been drawn.

The people who draw later never had an advantage or disadvantage. If the short straw has not been drawn by the time you take your turn, then the probability of you drawing it on your turn is higher than that of the person before you BUT the key is the "if" at the beginning: the chance of you having to take a turn in the first place (or, if you play the full game with secret draws, the chance of the short straw still being available to draw) is lower than that of the person in front.

For example, working with 6 people, the first person to draw clearly has a 1 in 6 chance of picking the short straw. After 3 draws, there is a 50% chance that the short straw has been drawn, and if it hasn't, there is a 1 in 3 chance of the 4th person picking the short straw. i.e. the 4th person has a 1/3 * 0.5 = 1/6th chance of picking the short straw.

What changes if you reveal straws and stop the game early is that *if* the 4th person has to draw, he knows at the time of drawing that he has a 1 in 3 chance, but there was a 1 in 2 chance that he didn't have to draw in the first place, so the overall probability remains 1/6th.

Oh this horse to drink wants to, but colleagues insist on instilling confusion: as the debate goes back & forth more random ideas are exuded and so it continues.

Thanks for the answers though – it's clear that the fair integrity of drawing straws is alive and well. Not sure how the odds are affected by the conveyor though… 😉

Oh, and rock isn't dead, but the odds of paper scissor stone are certainly affected when the fourth option (orbital ion cannon) is invoked.

……..reached the conclusion that drawing straws is actually an unfair way of selecting someone

What are you ………..some kind of socialist ?

Someone has to get the short straw………deal with it !

Have a go. If you are at work (and paid by the hour) draw straws 100 times and see if the 1st, 2nd, 3rd person etc each get the short straw the same number of times.