# When drawing straws, are you more likely to get the short straw if you pick first, or after several people have already picked?

When drawing straws, are you more likely to get the short straw if you pick first, or after several people have already picked?

In: Mathematics

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It depends. If everyone draws straws first and the everyone reveals at the same time, then it doesn’t matter when you pick yours.

If each individual draws their straw and then reveals it before the next person draws, then each later person has increased odds of getting the short straw IF they end up drawing a straw before the short straw has been revealed. However, there is also an increasing chance that the remaining people will not draw any straw at all. (so, actually, it still didn’t matter from a logical perspective, but it would be a lot more nerve wracking for you waiting last to pick as each person not to draw the short straw slowly increased your perception of how likely you were to draw it yourself.)

This is because odds are based on available knowledge. If the knowledge doesn’t change, the odds don’t change. If the knowledge changes, then the odds must change to reflect the new knowledge.

This is something you can demonstrate really neatly in the Monty Hall problem (three doors, two with things you don’t want, one with a thing you do want. If you pick one door without opening it, you have a 1:3 change that your door has the thing you want. next one of the remaining doors is opened to reveal a thing you don’t want- since there is a 2:3 chance that the door you picked had a thing you don’t want, but you’ve been shown new knowledge about a door you didn’t pick, you now know that there is a 2:3 chance that the remaining door actually has the thing you want and a 1:3 change that the door you initially picked has the thing you want, where before both had the same 1:3 chance.)

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I think a lot of what’s making this tougher to ELI5 has to do with the fact that the game stops when someone picks the short straw and the fact that one of the straws is “different”. That brings in a bunch of math that we can do without.

It’s far easier if we start with an identical, but simpler game:

All of the straws are different colors. There’s a sealed envelope that contains the name of one of the colors, chosen at random. Each person picks a straw, one after the other.

Now the envelope is opened, and whoever has the straw of that color is the loser.

It should be clear that this is the same game – you can think of the straw that matches the envelope as the “short straw”. But it may not be clear whether or not this game is “fair” or if the order matters. That’s okay.

Now let’s change it just a little bit. This time, everyone picks their colored straw, but the color for the envelope isn’t picked until after everyone has chosen. It is still picked in a fair and random way, but we just wait until after the straws to do it.

It should still be clear that the game is the same as the last one I described. It doesn’t matter whether the random color is chosen before or after the straws are chosen. Choosing the random color is unrelated to the players choosing straws.

But this game should be very simple to analyze: each player has an equal chance of having their color chosen. So each player has the same chance of getting the short straw.

Since the second game I described is equivalent to the first, and the first game I described is equivalent to drawing for the short straw, we can see that drawing for the short straw is indeed fair, and doesn’t depend on what order you go in.

As another response already pointed out, if the straw is pulled at any point, you are now guaranteed to NOT get the straw. Odds of getting the straw at the start of the pulling is 1 in how many straws exist. Odds of getting the straw if you go just prior to the final person are 50/50. But, that 50/50 is offset by the chances of not having to even bother pulling because the straw was already taken by someone else.

Since they illustrated with 4, I will illustrate with a prime number instead:

Say there are 7 people pulling at 7 straws:

* First man up: Chances to get the straw: 14%. Chances someone already got the straw: 0%. Net chances to pull the straw: 14% of 100%, or 14%.

* Second man up: Chances to get the straw: 17%. Chances someone already got the straw: 14%. Net chances to pull the straw: 17% of 86%, or 14%.

* Third man up: Chances to get the straw: 20%. Chances someone already got the straw: 29%. Net chances to pull the straw: 20% of 71%, or 14%.

* Fourth man up: Chances to get the straw: 25%. Chances someone already got the straw: 43%. Net chances to pull the straw: 25% of 56%, or 14%.

* Fifth man up: Chances to get the straw: 33%. Chances someone already got the straw: 57%. Net chances to pull the straw: 33% of 43%, or 14%.

* Sixth man up: Chances to get the straw: 50%. Chances someone already got the straw: 71%. Net chances to pull the straw: 50% of 29%, or 14%.

* Final man: Chances to get the straw: 100%. Chances someone already got the straw: 86%. Net chances to pull the straw: 100% of 14%, or 14%.

It works out that the odds for each person to get the straw are 14% (multiple their odds for that draw by their odds to have to draw at all).

It doesn’t matter.

The easiest way to think about this is that everyone who hasn’t already picked a straw has an equal chance of picking the short straw; the straw, after all, doesn’t care who picks it. It’s a straw. Straws are, notoriously, indifferent to people.

So say you’ve got five people deciding. Everyone has a 20% chance of pulling the short straw and ending the game. Person A pulls the straw, and success! Long straw. He’s safe. The game continues.

Now you’ve got a situation where there are four straws available, one of which is short. Everyone else has a 25% chance of pulling the short straw… but remember, there’s already been one round, which means there’s already a certain number of possible outcomes that have been taken off the board. (For example, if Person A pulls the short straw, there *is* no second round.) The odds of the game going to a second round is 80% (that is, the odds that Person A doesn’t pull the short straw). **Your overall odds, then, are 25% of that 80%… or, put another way, 20%.**

You can go down and down and down through the various [probability trees](https://www.mathsisfun.com/data/probability-tree-diagrams.html) of how the game might play itself out, but you’ll find that each person’s odds of picking a straw is 20% in any given round, *if* you don’t know the status of the straws pulled before.

That’s why it’s functionally equivalent to have everyone pull the straws one by one, or to have them pull the straws at the same time. The only difference is in how tense it gets when it’s down to two and you haven’t decided who’s paying for the pizza yet.

At the start, before anyone has picked, everyone has an equal chance of getting the short straw.

But as people pick and reveal their straw the probability changes now you know more information. If you’re last and no one has drawn the short straw yet then obviously you have a 100% chance of getting it yourself. All the other possibilities have been ruled out by this point.

On the flip side of this, you only have a 1 in <number of people> chance of getting to your turn with no one else having got it yet. So overall the probability is the same whether you pick first or last.

Neither, if it is a fair game (truly random, no cheating) each player has the same chance.

More info from a quick search:

Say you’ve got N peeps (people). The first person to draw a straw is the least likely to draw the short one (1/N) and the last to draw is the most likely (1/2). However! While the later people are more likely to draw the short straw, they’re also less likely to pull any straw since it’s more likely that the short straw has already been drawn. In movies they almost always draw every straw because of drama, but in practice, you draw until the short one shows up and then you stop.

It does not matter. Say there is 5 straws. The first one to pick have a 20% chance of picking the shortest one. Now there is 4 straws so the next person have a 25% chance of picking the shortest one. However that is assuming the first person did not already pick the short straw. And there is an 80% chance that he did not. And 80% of 25% is 20% which is exactly the same chance as the first one had. This is true for everyone involved. The last person have an 80% chance that someone already picked the shortest straw so even if he have a 100% chance of picking the shortest straw if that remains that is only a 20% chance.

You can think of the situation differently. What if instead of straws there was sealed envelopes. The order in which you pick your envelope does not matter as everyone is left with an envelope that might be the special one. The fact that you open the envelope just after you have picked it or after everyone have picked does not change its contents.