Many rolls, steady average

In the picture

1. 1Averages of a few rolls: widely scattered results
2. 2Averages of very many rolls: a narrow spike at the centre
3. 3The expected value: 3.5 for a fair die
4. 4Running average: big swings at first, then settling on the expected value

Examples

• A casino can’t predict a single spin of roulette, but over millions of bets its average profit per bet is almost exactly what the odds predict.
• An insurer can’t know which driver will crash, but across a million drivers the average claim cost is very predictable.

Quiz: Many rolls, steady average (Age 17)

1. 1

   Imagine many people each roll a fair die and work out their own average. Compared with people who roll it 10 times each, how will the averages of people who roll it 5,000 times each look?

   • More spread out, because more rolls give more chances for extreme results
   • About as spread out, just centred more exactly on 3.5
   • Much more tightly bunched around 3.5
2. 2

   A fair die has landed on low numbers several times in a row. What does the law of large numbers say about the next roll?

   • Nothing: the next roll is just as random as any other
   • A high number is now more likely, to balance out the low ones
   • A low number is more likely, because the die is on a streak
3. 3

   You roll a fair die again and again and update the running average after every roll. What should you expect to see?

   • It keeps swinging just as widely, however many rolls you make
   • It stays close to 3.5 from the very first few rolls
   • It swings a lot at first, then settles close to 3.5

The answers, each with its explanation, are in the app. This quiz is free there while the lesson is in the free week.