## Introduction to Lottery Wheels as Statistic Analyzers

Whenever we talk about lotteries, we often compare them to a game of roulette. In many ways roulette is a simple ancestor of lottery games because in both games there are a fixed number of possible outcomes, players select one or more outcomes against which to place bets, and there are techniques to bet on multiple outcomes.

The biggest difference between lotteries and roulette is that roulette provides a collection of side bets (“even,” “black,” “high,” etc.) so that players can quickly and easily bet on multiple outcomes.If roulette did not offer these options, a player wanting to bet on all the red numbers would have to place 18 individual bets. With some minor exceptions discussed below, lotteries do not offer side bets. Players who want to make side bets in lotteries have to bet on all the outcomes in which they are interested.

This is where the lottery wheels come in. Not to be confused with the physical wheel of a roulette table, a lottery wheel is a systematic method for selecting numbers in a lottery to create a customized side bet. A lottery wheeling system is one of the most popular methods, in which the player buys multiple tickets and arranges the lottery numbers on the tickets in such a way that they have higher chances of winning if the drawing results fall within a certain range of numbers.

Lottery wheels are particularly well known with lottery pools for two reasons: first, a lottery wheel ensures that every purchased ticket is unique; second, a lottery wheel can provide a limited assurance that the player will win some small amount.

The likelihood of winning the jackpot differs between various lotteries. The chances for the well-known US lotteries range from the absurdly high 1 in approximately 300 million for the double-pick multi-state lottery Mega Millions to the fairly good 1 in around 170,000 for the Pick-5, 31-number Wisconsin Lottery Badger 5. Wheeling systems are usually intended to offer a minimum number of wins if some of the drawn numbers are included in the set of the player's numbers.

Of all the techniques available for lottery players, lottery wheels are one of the most confusing and jargon-filled out there. Further, lottery wheels are often sold either as printable sheets, as software, or as part of a membership. This only encourages sellers to make lottery wheels seem unnecessarily arcane and complicated. Finally, many lottery wheels are described as providing a “guarantee;” however, what is actually guaranteed can be unclear.

So, let’s explore lottery wheels and learn how to put them to use for us.

Before we begin, recall that there are three types of lotteries.

“Number” or “digit” games such as Pick3 or Daily4 let players pick numbers from 000-999 or 0000-9999 respectively.These games usually allow some simple side bets such as a “box bet” or “wheel bet.” In both of these cases, the player wins if the numbers that they select appear in any order. For example, if the player picked 1-2-3, they would win if 2-1-3 was drawn. In these types of games, it is possible for a number to repeat (e.g. 5-5-5) in which case a box or wheel bet would give no advantage to the player. The main difference between a box and a wheel bet in a numbers game is the amount bet. Typically, a box bet only costs one dollar while a wheel bet costs one dollar for each possible outcome.

The second type of lottery is “lotto.” In lotto, there is a single machine that contains the numbered balls and all balls are drawn from that machine without replacement. This means that in a lotto game, a number can only appear once. Also, in this type of game, the order in which the balls are drawn doesn’t matter.

The final type of lottery is “bonus lotto.” This is a variation of lotto in which a second machine contains a second set of numbered balls. Typically, only one ball is drawn from this machine and to win the jackpot, players must correctly select the bonus ball and all of the lotto numbers. Games like Powerball and Mega Millions fall into this category.

Lottery wheels are specifically intended for use with lotto games. To give us some examples to work with, we will use Ohio’s Rolling Cash5 and Wisconsin’s Badger5. In both of these games, players select 5 numbers from a field of numbers. Rolling Cash5 has numbers from 1 to 39 and Badger5 has numbers from 1 to 31.

MATCH WIN/SHARE ODDS
5 of 5 \$10,000 * 1:169,911
4 of 5 \$50 1:1,308
3 of 5 \$2 1:53
2 of 5 \$1 1:7

## Ohio- Rolling Cash5

MATCH WIN/SHARE ODDS
All 5 of 5 winning numbers \$130,000 * 1 in 575,757
All 4 of 5 winning numbers \$300 1 in 3,387
All 3 of 5 winning numbers \$10 1 in 103
All 2 of 5 winning number \$1 1 in 10

The first step in building a wheel is to select a subset of numbers from within the field of numbers to be drawn. This subset can range in size from the number of balls drawn to the entire field. As the size of the subset increases, the cost of purchasing all the tickets needed to create the wheel grows exponentially. If you select a subset that has the same size as the number of balls drawn (5 in our examples), you are effectively making a single bet. If you select a subset that has the entire field of numbers, you are effectively attempting to cover the entire set of possible outcomes as Stefan Mandel did.

Before we move onto an example, now is a good time to explain what a wheel really does for a player. IF the results of the drawing are within the subset used to build the wheel, THEN the wheel increases (or in some special cases, ensures) the chances of a win. One of the ways that articles about lottery wheels can become confusing is that this condition is assumed. The creators of wheels may say that a wheel “guarantees” a certain result, but what they mean is IF the drawing results are within the subset used to build the wheel, then there is a “guarantee.”

Now, let’s introduce a very simple and inexpensive wheel. Suppose that for the Rolling Cash5 lottery, we want to create a wheel with a subset of six numbers. And for the sake of simplicity, we will use the numbers 1, 2, 3, 4, 5, and 6 for our subset. With such a small subset, it is easy to create a wheel with every combination(Table One).

Combinations Matches
Ball 1 Ball 2 Ball 3 Ball 4 Ball 5 Ball 6 IF 5 IF 4 IF 3 IF 2
First 2 3 4 5 6 4 3 2
Second 1 3 4 5 6 4 3 2
Third 1 2 4 5 6 4 3 2 2
Fourth 1 2 3 5 6 4 3 2 2
Fifth 1 2 3 4 6 4 3 2 2
Sixth 1 2 3 4 5 4 3 2 2

Table One: Full wheel for a Draw 5 Lotto using a subset of six numbers

As can be seen in Table One, there are six ways that these numbers can be combined. The wheel shown here is called a “full wheel” because it contains every possible combination. In a moment, we will introduce “abbreviated wheels” that contain fewer than the total possible combinations.

Also, by looking at the matches, we can see how lottery wheels impact the game. For example, suppose 1-2-3-4-5 is drawn so that all five numbers drawn are in the subset, then the player actually wins much more than just the jackpot. They also win five prizes for matching four of five numbers (a total of \$1,500 in Rolling Cash5). If four of the numbers drawn are in the subset (for example, 1-2-3-4-9), they get two prizes for matching four of five and four prizes for matching three of five.

This raises an important point about lottery wheels. The results that occur IF the balls drawn are in the subset is the same regardless of the total number of balls in the playing field. But the probability of balls being drawn from the subset are conditional on the total number of balls in the field. To illustrate this, observe that we could play the identical six combinations in both Rolling Cash5 and Badger5. The number and type of matches would be the same for both wheels. But because the fields of the games are different, the likelihood of matching say three numbers in the subset differs between the games.

As Table Two shows, the number of tickets required for a full wheel grows rapidly as the size of your subset grows. One way to offset this growing cost is to use less than the full wheel. This technique is called an abbreviated wheel.

WHEEL SUBSET NUMBER OF TICKETS
6 6
7 21
8 56
9 126
10 252
11 462
12 792
13 1,287
14 2,002
15 3,003

Table Two: Number of tickets required to purchase a full wheel

The primary disadvantage of an abbreviated wheel is that the “guarantee” of five numbers being drawn from within the player’s subset resulting in a match of five goes away. With an abbreviated wheel, the “guarantee” falls to “if four numbers are drawn from the subset, then a match of four is guaranteed” or “if four are drawn from the subset, then a match-three is guaranteed.” These “guarantees” are often shortened to “3 if 4”. There are countless abbreviated wheels.

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