How to Guess the Jelly Beans in a Jar: Mia's Method and Gary's 300
The short version: count jelly beans in a jar by multiplying, not guessing. Estimate the volume the beans fill, multiply by the share of space they actually occupy (randomly poured candies fill roughly 64% to 71% of a container; 65% is a sensible working figure for jelly beans), and divide by the volume of one bean, which you can measure by dropping 20 beans into water. For a half-gallon jar filled to the shoulder, about 1,750 ml × 0.65 ÷ 1.5 ml gives about 760. Small errors in each step partly cancel, which is why this method usually lands within 10% to 15%.
The school fair had a jar of jelly beans: half a gallon, filled to the shoulder, $1 a guess, closest wins the jar. Uncle Gary looked at it for about four seconds and wrote 300. "I counted the ones on the side," he said, "and doubled it, for the middle."
Cousin Kevin wrote 2,000, on the grounds that it was a big jar. Mia, who is fifteen and had been told she could have the jar if she won it, went home, came back an hour later with a measuring jug, and wrote 760.
Step 1: how much space do the beans fill?
The label said half a gallon: about 1,890 ml. The beans stopped at the shoulder, below the narrowing neck, so Mia knocked off a bit for the empty space at the top and called it 1,750 ml. Not exact, and doesn't need to be. Being 5% out here makes the final answer 5% out, not 500% out.
Step 2: how big is one bean?
This is where Mia used the measuring jug. She filled it with water to a line, dropped in 20 jelly beans from the same brand, and watched the water rise by about 30 ml. So one bean takes up about 1.5 ml. Twenty beans rather than one, because a single bean barely moves the line, and averaging over 20 smooths out the big ones and the small ones.
Step 3: how much of the jar is air?
Beans don't fill a jar like water. Poured in at random, they leave gaps. Physicists have measured this carefully: randomly packed spheres fill about 64% of a container, and in a 2004 study in Science, candies shaped like M&Ms packed more tightly, at about 68% to 71%. Jelly beans are lumpier than either, so Mia assumed 65%, and kept 60% to 70% as her range.
Step 4: multiply
- Space filled by beans: 1,750 ml × 0.65 = about 1,140 ml of actual bean.
- Number of beans: 1,140 ÷ 1.5 = about 760.
- With 60% or 70% packing instead: between about 700 and 820.
This kind of calculation is a Fermi estimate: break an impossible-looking question into a few steps you can estimate roughly, then multiply. Some errors push the answer up and others push it down, so they partly cancel. That's why Mia's rough steps beat Gary's precise count of the beans on the side, which only measured the outside of the jar and then guessed the inside.
What the fair guessed
There were 64 guesses. The middle guess, the median, was 650: closer than most individual guesses, as crowds often are, but still low. Big jars tend to be underestimated, because people count what they can see, and what they can see is the surface. The jar held 774 jelly beans. Mia was 14 out, about 2%. That's better than her method deserves: on another day, with a slightly different jar, she'd have been 10% out, and still won.
Jelly beans in the fair's jar: guesses and the real count
The jelly bean method
1. Volume of the beans. From the label or the jar's size, minus any empty space at the top.
2. Volume of one bean. Drop 20 into a jug of water and divide the rise by 20. No jug? A regular jelly bean is roughly 1 to 2 ml.
3. Packing. Multiply by about 0.65. Round sweets pack a bit tighter; long or flat ones vary more.
4. Sanity check. If your answer is far from the number you'd get by counting one layer and multiplying by the number of layers, one of your steps is wrong.
Where it costs you money
Big financial questions, such as how much a house will cost to run or what a business might be worth, look like jars: impossible to count, easy to guess wildly. Breaking them into steps you can check (price, rate, years, costs) gives a rough answer you can defend, and shows you which assumption matters most. A gut number like Gary's 300 can't be checked, which is exactly why it feels so confident.
The moral
Mia won the jar and gave a third of it to Gary, "for the middle". Gary has since announced that his method was right in principle and only wrong about the middle, which he now estimates is bigger than it looks. He has entered next year's guess in advance: 774.
The fair, the jar and the guesses are a worked example; the jar size, bean volume and 65% packing are assumptions. Packing figures for spheres and M&M-shaped candies are from Donev et al. (Science, 2004). Educational content, not financial advice.