The bell curve.

Step 1 of 5

Every peg is a coin toss

Balls fall from the hopper through a triangle of pegs. At every peg a ball bounces left or right, half and half, like a coin toss. At the bottom it lands in a bin, and the bin is set by how many times the ball bounced right: the leftmost bin is no rights at all, the rightmost is right at every row.

Here there are $4$⁠ rows of pegs, and so $5$⁠ bins. Every ball is four coin tosses in a row, just like the tree in the last Lesson.

Step 2 of 5

The middle bin

Only one path leads to each bin at the edge: right at every row, or left at every row. Many paths lead to the middle bin, two rights and two lefts, because the order can change: right-right-left-left, right-left-right-left, and so on.

All the paths are equally likely, so a bin’s chance is the number of paths into it out of all the paths. Look inside is set to Bell curve: the orange line shows how many balls to expect in each bin.

Step 3 of 5

The edges

Now there are $12$⁠ rows and $13$⁠ bins. To reach the rightmost bin, a ball has to bounce right at every one of the $12$⁠ rows. Predict how likely that is.

Step 4 of 5

Pegs that are not fair

Chance of bouncing right sets the chance at every peg. On average a ball bounces right at that share of the rows. With $n$⁠ rows and a chance $p$⁠ of bouncing right (as a fraction), the centre of the hump sits at bin

$n \times p$⁠

This is called the expected value: the number of rights you get on average. With fair pegs and $12$⁠ rows it is $12 \times 0.5 = 6$⁠, the middle bin. Change Chance of bouncing right so that the centre of the hump is at bin $9$⁠.

Step 5 of 5

The average is not the rule

Drag Rows of pegs up to $16$⁠ and look: the more rows, the smoother the bell curve the bars trace, a curve called the normal distribution. It turns up wherever many small, independent effects add up: people’s heights, measurement errors, the results of many tosses.

Now there are $10$⁠ rows: ten coin tosses. The average result is five rights, in the middle bin. But how often do you really get exactly $5$⁠ out of $10$⁠?