Does Turn Order Matter? Calculating the Odds of Going First

Every round of drawing straws brings the same debate: is it better to go first or last? The answer is it depends, and simple math shows why.

When slots match the number of players

Say there are four straws with one short, and four people draw in turn.

• The first person draws short with a probability of 1/4.
• The second person needs the first to pass (3/4) and then to draw short from the three left (1/3). Multiplied, that's 1/4.
• The third person's chance is 3/4 × 2/3 × 1/2 = 1/4.
• The fourth person's chance is 3/4 × 2/3 × 1/2 × 1 = 1/4.

Everyone has 25%. Later drawers face higher odds at their moment, but it's equally less likely the short straw is still there, and the two effects cancel exactly. Seeing what others drew may feel different, but the odds before the game are the same.

Here's an easy way to see it. Imagine all four people holding a straw and opening their hands at once. Changing the order in which hands open doesn't change what's in them.

When slots don't match the players

More slots than players changes things. In Russian Roulette, one of six chambers holds the flag. With four players pulling in rotation:

• Player 1 pulls chambers 1 and 5, for 2/6
• Player 2 pulls chambers 2 and 6, for 2/6
• Player 3 pulls chamber 3 only, for 1/6
• Player 4 pulls chamber 4 only, for 1/6

Here the first two players really are twice as exposed. This happens whenever the slots aren't a multiple of the players. With 2, 3 or 6 players, everyone pulls the same number of times and the gap disappears.

How to make it fair

Shuffling the order before starting solves it. With four players, anyone has a one-half chance of landing in the first two seats, so the odds of being caught are:

1/2 × 2/6 + 1/2 × 1/6 = 1/4

That's exactly the fair 25%. The seats still differ within a round, but seats are assigned at random, so nobody is disadvantaged when the game begins. All turn-based games on this site shuffle the order at the start.

When players outnumber slots

With seven players and six chambers, one person won't get a turn. With a shuffled order, each person has a 6/7 chance of getting a turn and a 1/6 chance of being caught on it, which comes to 1/7. It divides exactly by the number of players.

What about where you pick?

In Card Flip or Pop-Up Pirate, you might feel that a corner looks safer. But the trap is equally likely to be in any position. Enjoy the deliberating as much as you like, because it won't change the odds.