Published · 2026-10-25 · Last reviewed · 2026-10-25 · 6 min read

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Are shuffles, drawing lots and ladder games really fair?

To decide a speaking order or choose who cleans, we use shuffles, drawing lots and ladder games. Few people ask separately "is this method fair?"; mixing things up feels fair by itself. But the three are fair to different degrees. One is exactly fair, one is fair or unfair depending on how it is built, and one is badly skewed unless certain conditions hold. This page checks all three with counting and simulators. All calculation runs in your browser only.

1. Shuffling: it depends on how you mix

Shuffle three cards (0, 1, 2). There are 3! = 6 possible orders and a fair shuffle gives each 1/6 = 16.67%. A natural first idea is: "for each position, swap with some random position."

• Method A (swap with any position): for positions 0 to 2 in turn, swap that position's card with the card at a random position among all 3.
• Method B (Fisher–Yates): working from the back, swap each position's card with the card at a position among those up to and including it. The last position chooses among 3, the next among 2.

Method A looks plausible but is not fair. There are 3 × 3 × 3 = 27 swap paths and 6 final orders, and 27 does not divide by 6. Counting every path gives:

Some orders get 4/27 and some 5/27, straying above and below the fair 1/6 (= 4.5/27); the likeliest order is 25% more probable than the least likely. Method B has 3 × 2 × 1 = 6 paths for 6 orders, in one-to-one correspondence, so every order is exactly 1/6. Check it yourself.

A simulation has chance wobble, but the split between 18.5% and 14.8% in Method A shows clearly at 120,000 runs. What about 52 cards? Method A has 52⁵² paths against 52! orders, which do not divide evenly, and the bias does not go away as the deck grows. This site's number generator uses Method B (Fisher–Yates; see the code description in the random numbers article). It is also said that the common programming trick of "sorting with a random comparison" can be biased, but we did not test any specific implementation and make no judgement on it.

2. Drawing lots: fair whatever the order

A box holds n lots, one of which wins. n people draw one lot each in turn, without putting it back. Does the first or the last person have the advantage? Many feel the early drawers do, because a winning lot is more likely still to be in the box when they draw. The exact calculation says otherwise.

The chance that the k-th person draws the winning lot is found as follows: the first k − 1 people must all draw blanks, and then I must draw the winner.

(n−1)/n × (n−2)/(n−1) × … × (n−k+1)/(n−k+2) × 1/(n−k+1) = 1/n

The fractions cancel and give exactly 1/n at every position. An early drawer has a high chance that the winning lot is still there but a low chance (1/n) of picking it; a late drawer has fewer lots left and so a higher chance on that one pick. The two effects cancel exactly. With 5 people everyone has 20%. This is the mathematical basis for the fairness of drawing lots.

The conclusion has conditions, though. The lots must be indistinguishable by touch, well mixed before the draw, and what earlier drawers pulled must not change the mixed state seen by later ones. If the winning lot sits on top of the box, unmixed, the draw is not fair. Fairness is not a matter of the method but of how well mixed things are.

3. Ladder games: fair only with enough rungs

In a ladder game, rungs are placed at random between n vertical lines; starting from the top you follow a line down and cross to the neighbour whenever you meet a rung. Starts and finishes are paired one-to-one, so a permutation results. Whether it is fair depends on how many rungs there are.

Consider the extreme of very few rungs. With none, everyone ends on their own line. With a few, one can move only one line at a time, so someone starting at the far left tends to finish near the left. The simulator below takes the number of vertical lines and the number of rows where rungs may be placed, and shows the finishing distribution for the person who starts on line 1. In each row, whether to place a rung between neighbouring lines is decided by a coin flip, with the rule that rungs in one row cannot touch.

Results for four lines, varying the number of rows (400,000 simulated runs each). If fair, everything would be 25%.

With only 3 rows, the person starting on line 1 finishes on line 4 with 4.7% and on line 1 with 43.7%: a ninefold difference. As rows increase the difference shrinks and at 30 rows it is close to the fair 25%. Each crossing shuffles the position by one step at random, so enough repetitions wash out the starting point. How many rows real ladder games use and under what rule they place rungs differ from game to game and was not checked here. The results above are for "a simple rule placing rungs by coin flip in each row".

The lesson is that a ladder game is neither always fair nor always unfair; it depends on the number of rungs and the rule for placing them. A practical factor is the hand-drawn ladder: people placing rungs tend to create patterns rather than randomness, so where fairness matters it is better to hide the rungs and generate them at random.

Comparing the three

Use in daily life

• To fix an order, making number tickets, turning them over, mixing and drawing one at a time is the simplest and most exact by calculation.
• When shuffling in a program, use Fisher–Yates together with an unbiased function for random integers (rejection sampling).
• Treat a ladder game as fair only when there are enough rungs and nobody drew them by hand.

Limits

• The exact shuffle probabilities are a complete count for three cards; the size of the bias for larger decks depends on the structure.
• The ladder simulation is the result of the simple rule chosen here, and does not guarantee the rules of any real game or app.
• The simulators run on random numbers, so values differ a little from run to run.
• The fairness of drawing lots is an idealised calculation assuming a thoroughly mixed state.

Sources and verification

• LottoFlick generator source (Fisher–Yates shuffle) — computed by LottoFlick
• LottoFlick simulator code for this page (exact enumerations and simulations) — computed by LottoFlick

Related reading

• How Computers Make Random Numbers: Pseudo-Random, Secure and What LottoFlick Uses Randomness and fairness
• How a Lottery Draw Is Kept Fair: Machines, Observers and What Data Can Check Randomness and fairness
• The Birthday Problem: 23 People and a Coin-Flip Chance of a Shared Birthday Everyday probability
• Probability basics
• Number generator