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A crowd of dice

By David, 21 February 2017 Activity

Time to test your intuition. What’s better: one expert, or the wisdom of the crowds? Let’s let the dice decide.

You will need

• 10 cubic (six-sided) dice
• Two whiteboard markers (one black and one red)
• Pen and paper

What to do

Record how many ticks and crosses you get.

1. Take one dice. With the black marker, draw a cross on the 1 and 2 sides and a tick on the 3, 4, 5 and 6 sides.
2. Let the marker dry completely, and then roll the dice 20 times. On the sheet of paper, record how many crosses and how many ticks you got.
3. Take a second dice. Using the red marker, draw a cross on the 1 and a tick on the 2, 3, 4, 5 and 6 sides. Do you expect this dice to roll more ticks than the first one?

The black dice have two crosses. The red have only one.

4. Roll the second dice 20 times and record how many ticks and crosses you got. Was your prediction correct?
5. Take another 8 dice, and use the black marker to label them like the first one: crosses on the 1 and 2, and ticks on the 3, 4, 5 and 6.
6. Take all nine black-labelled dice and roll them together. If you get more ticks than crosses, record it as a tick. If you get more crosses than ticks, record it as a cross.

Roll all the dice together and record whether there are more ticks or crosses.

7. Before you repeat Step 6, take a moment to think. Will you record more ticks than the previous two experiments?
8. Repeat Step 6 until you have 20 results. Did you get the result you expected?
9. Clean the dice using soapy water. If it won’t come off, try using some isopropyl alcohol instead.

What’s happening?

Probability is a tricky thing. You might think the red-labelled dice will get more ticks than a handful of black-labelled dice. After all, it only has one cross and five ticks on it, while the black-labelled dice each have two crosses and only four ticks.

But as you add more and more dice, a second effect comes into play. When you roll one dice, getting a one or two is not particularly surprising. If you rolled 600 dice, you’d be pretty disappointed if more than half were ones or twos.

We can calculate exactly how many ones and twos you’d expect to roll using some simple probability. A dice has six sides, and, each side is equally likely to turn up. So after 600 throws, you’d expect 100 ones, 100 twos, 100 threes and so on.

Of course, you probably won’t get exactly 100 of each roll, but you’ll get something remarkably close. When we tried this experiment, we got totals of 90, 108, 90, 106, 98 and 108. This is known as the law of large numbers – when you do more and more trials on a probability experiment, the results will approach the expected value.

This activity uses significantly fewer than 600 dice, so it’s likely you’ll roll more crosses than ticks every now and then. On average, the handful should get more ticks than the red-labelled dice, but not by much. So you shouldn’t be surprised if the red dice does better!

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