Keno Sums

Introduction

This page shows the probability of any given sum of the number of 20 keno balls. For those not familiar with keno, it is a game in which the player bets on the draw of 20 balls, without replacement, numbered 1 to 80. I created this page when I noticed bets on the sum of the draw in Blitz Keno.

There are combin(80,20)= 3,535,316,142,212,174,320 ways to choose 20 items out of 80. The following table shows the number of combinations for any given possible total from 210 to 1,410, the probability of that total, and the probability of that total or less.

The average ball is 40.5, so the average sum is 20×40.5=810. As the table shows, the probability of getting exactly 810, which the mean, median, and mode, is 0.44%, or 1 in 227.46.

Programming Tips

My first attempt to get the combinations for any given total was by brute force, looping through all 3,535,316,142,212,174,320 ways to choose 20 balls out of 80 and taking a count of the sum of each. That program would have taken years to complete.

My second attempt was to look for a pattern. Perhaps a variant of the Fibonacci Sequence or line from a modified Pascal's Triangle. That was a noble attempt, but led to nothing.

My third attempt was at finding the generating function, but that would have been just as tedious as the first method. This can be done very easily in Mathematica, by the way, but I'm too cheap to buy it.

Finally, after some good programming tips from the math and computer geniuses at my forum, I made some programming shortcuts, which cut down the calculation time to under one second. Following is roughly the logic behind those shortcuts. This logic was inspired by code written by ZPP in this post on my forum.

As an example of this shortcut logic, the distribution of possible totals would be the same if the first three balls drawn were 1, 5, and 6 as if they were 2, 3, and 6. There is no need to loop through the other 18 balls twice. Likewise, the program never loops if the total of the remaining balls given the same number left and least possible ball has already been solved. This saves a HUGE amount of computer time.

I know that last statement is kind of heavy, so let me show you the critical line in my program:

a[k][level][m+t]+=a[m+1][level+1][t];

Here are the variable definitions:

Here is the full loop, after first filling in the 20th ball totals by hand.

	for (level=19; level>=1; level--) 
	{
		for (k=level; k<=60+level; k++) // lowest available ball
		{
			for (m=k; m<=60+level; m++) // level-th ball
			{
				for (t=0; t<=1410; t++)
				{
					a[k][level][m+t]+=a[m+1][level+1][t];
				}
			}
		}
	}

I hope that wasn't too confusing.

Internal Links

Lottery Sums — This same table, but based on a 6-49 lottery.

External Links

• Assume the balls will be drawn in order, from smallest to largest.
• Create an 81x21x1411 array. The first brackets will hold the smallest ball that can be drawn. The second the number of the next ball to be drawn (1 is the first, 20 is the last), and the third the total of all undrawn balls. The array will hold the count of the ways that total can be drawn, given the lowest possible ball and order of the next ball. The array could actually be 80x20x1410, but I prefer to not have to subtract one all the time to make the results fit.
• Initialize all elements of that array to zero.
• For the 20th ball, the number of combinations will be 1 for any total equal or greater than the smallest possible ball.
• Starting with the 19th ball, and going down to the first, loop through all possible balls you can still get. For the array element for that least possible ball, ball order, and possible total of all remaining balls add the array element for one number greater than the least possible ball, one level up, and the total remaining from that level up (not counting the number just drawn.
  • a = name of array
  • k = least ball possible to be drawn (remember they are drawn in order)
  • level = ball order (where 20 is the last drawn and 1 is the first)
  • m = last number just drawn for the given level
  • t = total of all remaining balls AFTER the ball (m) just drawn, but not including m.
  • WHAT IS KENO EXACT SCORE'S HOUSE EDGE? — First time this topic was raised on my forum.
  • SUM OF ALL 20 KENO BALLS — Second time this topic was raised on my forum.
  • Subset Sum Problem — How to solve this problem with a generating function (I don't recommend this method)

Written by: Michael Shackleford