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← The Last Card Problem

The Last Card Problem

Three short lessons for the grown-up beside a collector: why the first cards come fast, how long a whole set takes on average, and why the last few cost the most. Then roll a die until every face has turned up.

Spark & Whetstone · sparkwhetstone.org/learn/the-last-card-problem · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. Collecting is waiting

Roll a die and write down each face the first time it turns up. The first roll is always new. The second is new five times in six. By the time you have five faces, only one face in six is any use, so most rolls are repeats.

When something happens one time in n, you wait about n tries for it on average. That is why the end of a collection feels slow: every new card is rarer than the one before.

Question. You have seen five of the six faces. On average, how many more rolls until the last one turns up?

  1. 6 rolls
  2. 36 rolls
  3. 1 roll

Say it back. In your own words: why do the last faces of the die take longer to turn up than the first?

Lesson 2. The collector curve

Add up the waits. For six faces: 6/6 for the first, 6/5 for the second, then 6/4, 6/3, 6/2 and 6/1. That is 6 × (1 + ½ + ⅓ + ¼ + ⅕ + ⅙), about 14.7 rolls to see every face at least once.

The same sum works for any set of n equally likely items: n × (1 + ½ + … + 1/n). Mathematicians call it the coupon-collector problem, after cereal-box coupons. It grows a little faster than the size of the set, because the sum in the brackets keeps creeping up.

Question. A set has 100 different cards, all equally likely, one random card per pack. About how many packs does a whole set take on average?

  1. About 520
  2. About 200
  3. About 100

Say it back. In your own words: why does a set of 100 cards take far more than 100 packs?

Lesson 3. Why the last card costs most

Split the collection in half. The first half comes quickly, because almost every pack holds something new. The second half is where nearly all the waiting lives.

That is why swapping duplicates with a friend, or buying the last few cards as singles, saves so much: the slow end is the part a swap replaces. It is also why a sticker album can feel nearly done for a long time.

Question. Of the roughly 519 packs a 100-card set takes on average, about how many go on the first 50 cards?

  1. About 69
  2. About 450
  3. About 260

Say it back. In your own words: why does swapping duplicates help most near the end of a collection?

At the table: Roll until every face appears

Players: Two or more, ages 6 and up. You'll need: One die each, a pencil and a strip of paper with 1 to 6 written along it.

  1. Before rolling, everyone guesses how many rolls it will take to see all six faces.
  2. Roll and tick each face the first time it turns up; count every roll, repeats too.
  3. Write down the count when the last face appears, then play twice more.
  4. Add up everyone’s counts, divide by the number of games, and compare that average with about 14.7, the long-run average.

Answers

  1. 6 rolls — The missing face comes up one roll in six, so you wait 6 rolls on average. Some days it is the next roll; some days it takes twenty.
  2. About 520 — 1 + ½ + … + 1/100 is about 5.19, so the set takes about 519 packs on average, more than five packs per card. Real sets with rare cards take longer still.
  3. About 69 — The first 50 cards take about 69 packs; the last 50 take about 450. Most of the cost is at the end.