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