How Many Spins Until Every Student Gets a Turn?

The short answer

In a class of 25, with winners left on the wheel (the student picker's Take each student off the wheel once picked turned off), the wheel needs 95.4 spins on average before every student has been picked at least once. That is 3.8 spins per student, so about 70 of those calls go to students who have already had a turn.

The average hides a wide spread. Half of all classes are covered by spin 90, 1 class in 10 needs 135 spins or more, and 1 in 100 needs 192 or more. Only 1.5% of classes have everyone picked within 50 spins, two per student, and only 27.3% within 75.

These figures are exact, not estimates from a sample. The average comes from the formula in the next section, and the median and percentiles from an exact calculation of the odds after every spin. We also ran 480,000 simulated classes through random.js, the file that picks every winner on Wheel Cue, and the results agreed with the math within the expected simulation noise. Across 80,000 simulated classes of 25, the average was 95.5 spins against the exact 95.4; the luckiest class needed 31 spins and the unluckiest 357. All of the spins on this page are simulated. None of them come from people using the site.

Why the last few names take so long

This is a classic problem in probability, usually called the coupon collector's problem: how many random draws until you have collected every item in a set?

The first spin always finds someone new. The second finds someone new 24 times out of 25. But once 24 students have had a turn, each spin has only a 1 in 25 chance of landing on the one student left, so that last name takes 25 spins on average by itself. Adding up every step gives the exact average: 25 × (1 + 1/2 + 1/3 + … + 1/25) = 95.4.

The cost is lopsided. The first 13 of 25 students, just over half the class, take 17.8 spins on average to reach. The last five take 57.1, which is 60% of the whole wait. On a real wheel that feels like the same few names coming up again and again while one or two students never hear theirs.

Because the average per student is 1 + 1/2 + … + 1/n, it keeps rising with class size, slowly. A bigger class does not just take longer in total. It takes longer per student.

For a class of 25, the chance that everyone has been called is 64% by spin 100 and 95% by spin 150. By the time the last student is finally called, the student who came up most often has been called 8.0 times on average.

After one spin per student, more than a third of the class is still waiting

Suppose you spin exactly 25 times for a class of 25: one spin per student, which feels like a fair share. With winners left on the wheel, the exact expected result is:

• 9.0 students never picked, or 36.0% of the class. The formula is 25 × (24/25)^25.
• 6.6 students picked twice or more, taking the turns the others missed.
• A 93.3% chance that some student was picked three or more times.
• The busiest student picked 3.4 times on average.

Luck barely helps. In 99.9% of the 80,000 simulated classes of 25, at least five students were still waiting after 25 spins, and even the luckiest class had 2 left.

The share left out barely depends on class size. It is 34.9% for a class of 10 and 36.3% for a class of 35, and as the class grows it approaches 1/e, or 36.8%. For any class of ten or more, that is more than a third of the room still waiting after one spin per student.

Try it with your own class size below. The simulation spins until everyone has been picked, many times over, and compares the result with the exact average.

One lesson and one week of random calls

A lesson usually holds a handful of random calls, not a hundred. Here is what that looks like in a class of 25, with every winner left on the wheel.

One lesson, 10 calls

• Any given student has a 66.5% chance of not being called at all, about two in three.
• On average 16.6 of the 25 students are not called, so the 10 calls reach only 8.4 different students.
• There is an 87.6% chance that at least one student is called twice or more (1.5 students on average), and a 15.1% chance that someone is called three or more times.

Most of that is fine. Nobody expects every student to speak in every lesson, and a repeat is a normal part of independent draws, not a sign of a broken wheel.

One week, 5 lessons of 10 calls

Over 50 calls a fair share would be exactly 2 calls per student. With replacement, it comes out like this:

• Any given student has a 13.0% chance of going the whole week without being called, about 1 in 8. The formula is (24/25)^50.
• On average 3.2 students are never called all week, and there is a 98.5% chance that at least one student is.
• 10.0 students on average, 40% of the class, get one call or none.
• 3.5 students on average get four calls or more, and the busiest student gets 5.2 calls on average against a fair share of 2.

So across a week, leaving winners on the wheel almost guarantees at least one student who is never asked.

What changes when winners come off the wheel

With Remove the winner after each spin on, each name leaves the wheel as soon as it is drawn. A round of 25 is exactly 25 calls, one per student, and no one can be called twice in it. Because Wheel Cue will not spin with fewer than two entries, the round takes 24 spins: the last name left simply goes last. Refill the wheel and the next round starts.

Over the same week of 50 calls, that means:

• Every student is called exactly 2 times. 50 calls make 2 full rounds of 25. With replacement, by comparison, 10.0 students on average get one call or none.
• Nobody waits more than 49 calls between turns. The longest possible gap is from first in one round to last in the next. With replacement, after any turn there is a 13.0% chance that the same student's next turn is more than 50 calls away, and no upper limit at all.
• The average wait is the same. In both modes a student waits 25 calls between turns on average. Removing winners does not change the rate, only the spread around it.
• The order is still random. Each spin is a fair draw from the names left, so every student is equally likely to be called first, last or anywhere in between.

These follow from the procedure itself, so with the whole class on the wheel they hold every week, not just on average. The guide to selection with and without replacement explains why each remaining name's odds rise as the wheel shrinks, and why that is fairer over a round, not less.

What to do with this in class

The numbers point to one decision: be clear about what each spin is for.

For sharing turns, take winners off

On the random student picker, Take each student off the wheel once picked is on by default. On the name picker the same option, Remove the winner after each spin, is off by default, so switch it on in the wheel options. The strip under the wheel then shows the odds rising from 1 in 25 as names come off, which is worth putting on the board with Fullscreen so the class can see the shrinking pool.

At 10 calls a lesson, a round of 25 lasts 2.5 lessons, so every round runs across more than one lesson: press Save before you close the tab and load the round in progress next lesson. That changes what Reset can do: it puts back the list as it stood at the first spin after you last loaded or edited it. If nobody edits the list during the round, one press of Reset at the end brings the whole class back. Deleting an absent student, adding a late arrival or fixing a typo starts that count again, and so does loading a saved round in progress, so keep the full class saved as its own wheel and load it to start the next round. The guide to picking students randomly has the full routine, and How it works explains Reset and the results log.

The exact shares above assume the whole class is on the wheel. A student you delete for an absence misses that round's turn, so note the name and call on them yourself when they are back.

With Weighted choices on and winners removed, a weighted line such as Maya *3 still leaves the wheel after one win: the weight makes that student more likely to come up earlier in the round, never gives an extra turn, and the strip shows total weight instead of 1-in-25 odds. With winners left on, a weight does mean more turns, which the weighted picker guide explains.

If you leave winners on, keep a record

The results panel (Who has been picked on the student picker, Picked names on the name picker) shows the latest spins, newest first, and Copy exports every spin since the page was opened or the log was last cleared as a table, oldest first. The log is cleared when the tab closes or reloads, so paste it into one spreadsheet at the end of each lesson. With 50 calls a week, expect about 3.2 students to be missed: by Friday, the names missing from that sheet are the students to call on yourself.

When to leave winners on the wheel

Replacement has one real advantage: nobody is ever safe. The usual worry with removal is that a student who has just answered knows they will not be called again until the round is over, and may stop paying attention. With every name on the wheel, each spin gives every student the same 1 in 25 chance, including the one who answered a minute ago.

That is worth having for quick checks of understanding, where the point is that everyone thinks before the spin. It is a poor way to share turns. Use replacement when any single call does not matter much and you are not tracking who has spoken, and use removal whenever the turn itself is the point: reading aloud, presenting, sharing answers, taking a classroom job.

A middle path keeps both benefits: run rounds with winners removed for the main questions, and ask follow-ups to anyone, including students who have already had a turn. Every student still gets exactly one turn per round, and no student can assume they are off the hook. On a wheel with removal off, Remove & spin in the result window takes off just that winner and spins again, so you can also decide call by call.

The same reasoning holds outside a classroom. A club leader taking turns, a coach rotating who leads warm-ups, or a game night rotating who goes first each round all face the same math. If the turns matter, take winners off and refill the wheel.