Lesson 6 of 11

When the two disagree

8 min

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A die is rolled 15 times. A 4 shows 5 times. Is the die fair? A fair die is equally likely to show each face.

We cannot tell yet. A fair die can show five 4s in 15 rolls.

0.33 from 15 rolls, 0.17 from the die's six faces. A gap like this can happen with few rolls.

Two figures, two questions

Real rolls give 5/15 = 1/3, about 0.33. The die's six equal faces give 1/6, about 0.17. The two are not the same number.

The observed chance reports what happened. The worked-out chance reports what the set-up allows. With more rolls, the two chances usually get closer.

Experimental — the observed chance, from real rolls. Theoretical — the worked-out chance, from the die's faces.

Two figures for a 4 on a die
Figure	Comes from	Value
Experimental	15 rolls	0.33
Theoretical	the die's 6 faces	0.17

Roll the die 10000 times. The gap is usually much smaller. It still does not have to be zero.

Rule: Real rolls give one figure. The die's faces give another. With few rolls a gap is normal. It does not prove the die is unfair.

Drag the rolls. Watch the gap.

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