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Sampling With Replacement vs Without Replacement Made Clear
Compare sampling with replacement vs without replacement using ten numbered tickets, a changing-pool table and settings for three random numbers per batch.
Sampling with replacement vs without replacement comes down to what happens after each pick. Put the chosen ticket back, and it can appear again. Keep it out, and the next pick comes from a smaller pool. That one decision changes which sequences are possible.
Use ten tickets numbered 1 through 10 and draw three. The worked sequences below are teaching examples, not recorded random results. You can follow them with paper tickets or try the corresponding settings in the FunBox random number generator.
Use ten numbered tickets to define replacement
Make each ticket the same size, give it a different number and mix the ten thoroughly. Each available ticket should have an equal chance at each pick. Record the number before returning it to the pool or setting it aside. Replacement describes that return step.
With replacement, all ten tickets are available before every draw. Without replacement, the pool drops from ten to nine after the first draw, then to eight after the second. The table follows the same starting conditions under both rules.
| Draw | With replacement | Without replacement |
|---|---|---|
| First | 10 tickets, numbered 1 through 10 | 10 tickets, numbered 1 through 10 |
| Second, after picking 4 | 10 tickets, including 4 | 9 tickets, excluding 4 |
| Third, after also picking 9 | 10 tickets, including 4 and 9 | 8 tickets, excluding 4 and 9 |
Follow three draws with replacement
Suppose the recorded sequence is 4, 9, 4. Draw 4, write it down, return it and mix again. Draw 9 and repeat that process. On the third draw, 4 is available again, so the sequence is allowed. Every draw still offers ten choices.
The chance of 4 on the next draw stays 1/10, even after 4 has just appeared. OpenStax explains independent draws through this unchanged probability. For this ticket model, returning and thoroughly mixing each ticket keeps later choices independent of earlier results.
The exact ordered sequence 4, 9, 4 has probability 1/10 × 1/10 × 1/10 = 1/1,000. So does 4, 9, 2. There are 1,000 possible ordered three-draw sequences, each equally likely with equal, independent draws. A repeat alone gives you no reason to discard the batch.
Replacement permits repeats without requiring them. Three different numbers can appear too. Seeing only distinct values in one batch therefore tells you little about which rule produced it. Check the settings or the physical procedure.
Follow three draws without replacement
Now use 4, 9, 2. After recording 4, leave that ticket outside the pool. Draw 9 from the remaining nine and set it aside too. The third draw has eight eligible tickets, including 2. Drawing 4 again during this sequence is impossible because its ticket is already out.
Conditional on those earlier picks, the chance of 9 on the second draw is 1/9, and the chance of 2 on the third is 1/8. For this exact ordered sequence, multiply 1/10 × 1/9 × 1/8 = 1/720. The shrinking pool makes the draws dependent.
Before any ticket is drawn, each number still has a 1/10 chance of occupying the second position. The 1/9 figure applies after you know which different ticket came first. Keeping that condition attached to the fraction prevents a common confusion about the shrinking pool.
There are 720 possible ordered sequences of three distinct tickets. For an unordered sample, each three-ticket set can appear in six orders, leaving 120 equally likely sets. Keep the ordered version when the positions assign first, second and third turns. Use the set when only membership matters.
Match the examples to the number generator settings
Open the generator and set Minimum to 1, Maximum to 10 and Quantity to 3. Both endpoints count, so this range contains ten integers. Leave Sort smallest to largest unchecked while comparing the examples, so you can read the generated order.
For the replacement model, leave No repeats in this batch unchecked and press Generate numbers. For random sampling without duplicates, check that option before generating. Your numbers will vary; the examples above describe allowed sequences rather than targets to reproduce.
| Setting | With replacement | Without replacement |
|---|---|---|
| Minimum / Maximum | 1 / 10 | 1 / 10 |
| Quantity | 3 | 3 |
| No repeats in this batch | Unchecked | Checked |
| Sort smallest to largest | Unchecked to preserve order | Unchecked to preserve order |
Keep separate batches separate from one long draw
Can the same number appear in the next batch? Yes. FunBox starts each batch from the full range. If a no-repeat batch contains 4, 9 and 2, all three become eligible again when you generate another batch. The option excludes duplicates within that batch only.
For six different numbers, request Quantity 6 in one no-repeat batch. Two batches of three can overlap. For a complete ten-ticket turn order, request all ten at once and copy the result before assigning turns. Generating a fresh batch starts a fresh order.
A range of 1 through 10 cannot supply eleven distinct numbers. Reduce the quantity or enlarge the range. FunBox allows up to 100 numbers per batch, so a larger sequence may need another method that explicitly retains the remaining pool.
Does sorting change which values are selected?
Sorting rearranges the selected values. It keeps the same numbers and any repeats already present. A draw recorded as 4, 9, 2 would display as 2, 4, 9 after sorting. That is convenient for checking membership, but it loses the original turn order.
If you are assigning first turn, leave sorting off. If you only need three distinct ticket numbers, sorting can make them easier to read. Copy the displayed result when you need a record, because refreshing the page clears the current batch.
For choosing a person by name, our Random Picker vs Wheel guide compares two other workflows. For this numbered example, try one batch with no repeats, then decide whether your next activity needs a fresh full pool or a continuing sequence.