## The Pinball Game

The
picture to the right shows a type of pinball machine that you can
build yourself. You will need 10 finishing nails, 5 small cups, a
wooden board and a pinball (marble). Nail the nails part way into
the board in the triangular pattern shown, with one nail in the
top row, two in the second, three in the third and so on, and
with enough space for the pinball to fit between the nails.

To operate the machine tilt the board at a slight angle and
release the pinball so that it hits the top nail dead center. The
pinball will be deflected either left or right with equal
probability by the first nail. It will then continue falling and
hit one of the nails in the second row and be deflected either
left or right around that nail with equal probability. (Some
experimentation will be required with the tilt of the board to
get the probabilities of going left and right to be equal.)

The result is that the pinball follows a random path, deflecting
off one pin in each of the four rows of pins, and ending up in
one of the cups at the bottom. The various possible paths are
shown by the gray lines and one particular path is shown by the
red line. We will describe this path using the notation
"

*LRLL*" meaning "

*deflection to the left around the
first pin, then deflection right around the pin in the second
row, then deflection left around the third and fourth
pins*".

**Question: How many different paths are there through the
pinball machine and what are they?**
**Answer:** Click here
for an explanation. There are 16 different paths through the
machine. The list of paths is:

*LLLL LLLR LLRL LLRR*

LRLL LRLR LRRL LRRR

RLLL RLLR RLRL RLRR

RRLL RRLR RRRL RRRR

**Question: How many paths are there that end up in any
given bin?**
To answer this question sort the list of the 16 different paths
according to how many

*R*'s each path contains:

Number of *R*'s = 0: { *LLLL* }

Number of *R*'s = 1: { *LLLR LLRL LRLL RLLL*
}

Number of *R*'s = 2: { *LLRR LRLR LRRL RLLR RLRL
RRLL* }

Number of *R*'s = 3: { *LRRR RLRR RRLR RRRL*
}

Number of *R*'s = 4: { *RRRR* }

If the number of

*R*'s =
0 then the pinball ends up in the first (leftmost) bin, (1
path)

If the number of

*R*'s = 1 then the pinball ends up in the
second bin, (4 paths)

If the number of

*R*'s = 2 then the pinball ends up in the
third bin, (6 paths)

If the number of

*R*'s = 3 then the pinball ends up in the
fourth bin, (4 paths)

If the number of

*R*'s = 4 then the pinball ends up in the
last (rightmost) bin, (1 path).

The result is as shown in the picture to the right.

**Question: What is the probability that the
pinball will end up in any given bin?**
There are 16 paths and each path is equally likely so each path
has a 1 in 16 chance of being followed. Thus the probabilities
are:

- probability = 1/16 to end up in first (leftmost) bin
- probability = 4/16 to end up in second bin
- probability = 6/16 to end up in third bin
- probability = 4/16 to end up in fourth bin
- probability = 1/16 to end up in fifth (rightmost) bin

A histogram of these values is shown
to the right. If we drop many pinballs through the machine and
let them pile up in the bins then over the long run they will be
distributed as shown in the histogram to the right. This is an
example of the

**binomial distribution** which is
studied in probability. The

**bell curve** or

**normal distribution** is based on this
distribution.

## Pascal's Triangle

The object to the right is known as

**Pascal's Triangle**. (Blaise Pascal, its
discoverer, was born in France and died in 1662 at the age of 39.
The Pascal programming language is named after him.)

Pascal's triangle is very useful for analysing the pinball
machine. Pascal's triangle also pops up in a variety of other
seemingly unrelated areas. First we mention that the triangle
continues on forever and we have only shown the first 5 rows. Can
you see the pattern and guess what the next row of numbers
is?

**Click here to
continue.**

Written by Eric Hiob,
Saturday, December 28, 1996 - 2:53:44 PM