# The Clown Always Wins

I have been teaching mathematics in an Australian High School since 1982, and I am a contributing author to mathematics text books.

I knew summer was here when the carnival arrived. Even with depleted funds, my friends and I still enjoyed balmy days meandering from one side-show attraction to another, often to the accompaniment of muffled music drifting from speakers perched on anchored poles. When our pocket money allowed it, we observed the panorama atop the Ferris wheel, deciding which sideshow attractions we will investigate.

Reminiscing about carnivals always kindles memories of playing -and consistently losing- the laughing clown game. Now, many years later and with mathematics on my side, I believe I know the reasons.

## How the game is played

There are nine slots numbered 2, 4, 5, 1, 3, 6, 4, 2, 3. Five ping-pong balls are dropped one at a time inside the mouth of the moving clown, and each ball lands on one of the nine numbers.

Below are three combinations of how the balls can be distributed.

The total of the five numbers determines what prize (if any) is won.

The designers of the game must be astute, trained mathematicians! Otherwise, how else is it that the sum required for a major prize seldom appears on the list of winning combinations?

To see why this is the case, we need to examine the game using probability theory.

Firstly, we will use the multiplication principle to calculate the total number of ways the ping-pong balls can fall on the nine slots.

Think of five squares, with each square representing the nine different numbers the ball can land on.

The multiplication principle states that the total number of ways is the product of the five numbers.

There are 59,049 ways the five balls can distribute themselves across the nine numbers.

## Combinations

The second thing we must do is to find all possible combinations of the total of the numbers the balls land on, where each total is a number from 5 to 30.

Let's look in some detail as to how to calculate combinations for totals of 5, 6 and 7.

Total of 5

Scroll to Continue

There is only one was a total of five can occur, and this is when each ball lands on 1.

Note: The balls are shown in different colours to indicate order, which is important.

Total of 6

There are ten different combinations that produce a total of 6.

Note that the order of the balls is now taken into account.

Listing or displaying combinations can be a tedious process. Applying combinatorial theory can greatly assist in making calculations efficient.

A useful formula in combinatorics provides the number of ways groups of objects can be selected from the main group, given that some objects are identical.

To obtain a total of 6, one combination is 1, 1, 1, 1, 2.

This means we use N = 5 (5 balls), n1 = 4 (the number 1 is repeated 4 times), n2 = 1 (there is one 2).

Now, in our group of nine numbers, 2, 4, 5, 1, 3, 6, 4, 2, 3, the number 2 appears twice. This means the total number of combinations is 2 x 5 = 10.

Total of 7

There are 50 different combinations where the sum of the balls is 7.

One combination is 1, 1, 1, 2, 2.

Use N = 5 (5 balls), n1 = 3 (the number 1 is repeated 3 times), n2 = 2 (the number 2 is repeated twice.)

Each of the numbers 2 appears twice in the nine numbers, so the total number of combinations is 10 x 2 x 2 = 40.

Another combination that sums to 7 is 1, 1, 1, 1, 3.

Use N = 5 (5 balls), n1 = 1 (the number 1 is repeated 4 times), n2 = 1 (there is only one of number 3).

The number 3 appears twice, so the number of ways for this combination is 2 x 5 = 10. Hence, the total number of combinations 40 + 10 = 50.

The number of combinations for the remaining sums up to 30 can also be found in this way. The table below summarises the number of combinations and the percentage equivalent for all totals up to 30.

Notice a near symmetry in the number of combinations (column 2). They start at 1, increase to 6840 and then decrease back to 1.

The graph of these results produces the normal distribution curve.

The graph shows that 95% of the combinations come from ping pong totals that lie in the interval approximately 10 to 23. Approximately 2.5% of the combinations are below a total of 10 and 2.5% above 23.

## Winning combinations

Winning numbers typically displayed with the game are shown below.

Now we can see how the game’s creators decided on the prize distribution.

Looking at the graph, it seems that total scores from about 13 to 20 (about 85%) are relatively easy to get, so these scores are not given ‘winning’ status.

Scores starting at 13 and progressively getting lower are harder to achieve, as are scores starting at 21 and increasing to 30.

A possible distribution of prizes is shown below.

Only now do I understand why I got stuck with a lousy pencil as a prize each time.

I also now understand why clowns are always laughing. You can never beat them at their own game.