The Galton board: order from random bounces
Name: Date:
Before the film: predict
1,000 balls fall through 10 rows of pegs. At every peg a ball goes left or right with equal chance. Sketch where you think they pile up.
Starter
A board has 3 rows. Draw every left/right path a ball can take. How many are there?
How many of the 8 paths end in each of the 4 bins?
What is the chance that a ball lands in one of the two middle bins?
Core
With 10 rows, how many different paths can one ball take?
How many of them end in the middle bin (exactly 5 bounces to the right)?
What is the chance of the middle bin?
What is the chance of one of the two outer bins?
Out of 1,000 balls, about how many land in bins 4, 5 or 6?
Challenge
Every peg now sends 60 % of balls to the right. Which bin of the 10 is most likely, and how likely?
With an even number of rows n, the middle bin gets C(n, n/2) ÷ 2ⁿ of the balls. From how many rows is that below 10 %?