Now noted mathematical analyst Joseph E. Granville, creator of successful stock market strategies used by thousands, has directed the enormous power of his analytical mind to the game of Bingo. After years of painstaking research, he has developed proven strategies that give you a clear competitive edge so that you can actually beat your luck at Bingo.

Granville's techniques are so simple anyone can use them. There's no complicated figuring, no giant mental calculations to be done. Granville lays out the simple step-by-step procedures for you to follow which automatically turn any game of Bingo you play in your favor.

Sound impossible? It isn't. Extensive study of thousands of games has led Granville to the inescapable conclusion that every bingo game halls follows definite patterns¡ patterns the average player is completely unaware of. By utilizing these patterns, Granville had discovered how to beat the odds at Bingo. Now you can too.

Naturally, the heart of any winning Bingo system is card selection. Granville has isolated crucial relationships between winning Bingo numbers and the master board. He shows you how to use these simple and proven truths to select a greater number of winning cards. Most methods players use to select their cards are completely backwards, Granville found. Players are working against themselves without even realizing it.

Even in games where you can't select your cards, there are ways to beat the odds and come up a winner. For instance, most Bingo enthusiasts play several cards a game to improve their chances of winning. But does this really work? No, says Granville! The startling truth is that you can actually improve your chances of winning big by playing fewer cards in many cases. Granville proves it! Curious? Read on to find out how fewer cards can be better.

So why trust to luck when you play Bingo? You can make the game pay you to play. If you're honestly serious about becoming a systematic winner at Bingo, here is an idea that you can use today.

The most natural reaction to advancing a serious theory designed to improve the chances of winning at bingo is encountered when confronting those who do not believe that such a sound theory is possible. The usual reaction to those who might devise various bingo "systems" is that it is all pure fantasy. They will tell you that nobody knows what balls are going to come out of the machine and that the game is totally one of luck. While it may appear at first glance difficult to counter such a reaction, the solid structure of mathematical probability is capable of destroying the argument. The key to beating the bingo game halls lies in a clear understanding of the word random. Our typical critic will agree that the colored balls being drawn from a machine are popping out at random. Now, having a common agreement on this fact, the next step is simply to show such critics that there is far more to the word random than first meets the eye.

As every player knows, there are 75 balls in the machine, numbered from 1 to 75. The probability of any ball coming up on the first draw is exactly equal, 1 in 75, written as 1/75. Since the probabilities are equal, we call this a uniform distribution. Random for s H numbers drawn from a uniform distribution fall into predictable patterns governed by the laws of probability. Therein lies the answer to transforming the otherwise hopeless problem into a series of systematic solutions which will determine the best selection of bingo cards. Granted that the balls come out of the machine at random, then three things must have a strong tendency to occur.

1. There must be an equal number of numbers ending in 1's, 2's, 3's, 4's etc.

2. Odd and even numbers must tend to balance.

3. High and low numbers must tend to balance.

Those are the three accepted tests for randomness. Unless the distribution meets those tests it is said that there is a bias and the distribution is not random. We can add a fourth test for randomness which has a peculiarly effective application at beating the bingo game halls.

This fourth test is best described by the English statistician L. H. C. Tippett in his book, Sampling.- "As a random sample is increased in size, it gives a result that comes closer and closer to the population value." Translated into simple everyday language, the bingo master board of 75 numbers constitutes the "population". The average number in that population is the average of the entire 75 numbers. Going from 1 to 75, the average number on the bingo board is 38. The first few numbers called in a bingo game halls may or may not average 38, but it is certain that as the game progresses the average of the numbers called will steadily approach 38. The author will wager that not one in ten players is aware of this statistical fact. So then, when bingo numbers are being called, the entire game (which consists of an average of 12 calls) is a sampling of the entire population and the larger the sample the closer the numbers will average to 38. Obviously this fact will play a key role in the strategic selection of bingo cards.

The next time you play bingo, note very carefully an amazing characteristic relating to the first ten numbers flashed on the master board. With very few exceptions, you will note that a preponderance of the numbers have different digit endings! The average bingo player, putting all the attention on the cards rather than the master board, would tend to overlook this, the most important single characteristic of the first ten numbers called in any bingo game halls. Since most regular games last for about ten to twelve calls or less, you will vastly improve your chances of selecting a winning card by concentrating on numbers having different digit endings.

PROBABILITY PREDICTS DIFFERENT DIGIT ENDINGS

The reason behind this important piece of information goes back to the first characteristic of drawing numbers at random from a uniform distribution. The first expectation would be that there would be an equal number of numbers ending in 1's, 2's, 3's, 4's etc. Since we are only concerned with the first ten or twelve numbers to be called, not enough balls have been drawn to expect more than a minimum of digit pairs. The laws governing a sample drawing of ten balls out of seventy five would show a strong tendency toward there being one ball with a number ending in 1, another ending in 2, another ending in 3, etc. until most of the ten digit endings are represented. The law is derived from simple probability. If the first number called in a game is N-31 then all the probabilities are increased on the next draw that the second number will not end with the digit 1 simply because there are more balls having different digit endings than there are balls left with numbers ending in 1. If the next number is G-56 then the probabilities are increased that the next number will not end in 1 or 6. For the first six numbers called in a game the probabilities are clearly in favor of all having different digit endings. From the seventh number on the probabilities favor pairing up one or more of the ending digits. This then accounts for the actual experience wherein it is shown that 60% of the first ten numbers called in any bingo game halls will tend to have different digit endings.

To validate the writer's assumption that there is a natural tendency first toward numbers having different digit endings, 49-game series was reviewed and the first ten numbers in each game is rated percentage wise for different digit endings.

Every-bingo card consists of 24 numbers and the free spot in the center. Those 24 numbers occupy 16 strategic squares, the remaining numbers covering the dead squares. The clear majority of all winning-bingo combinations consist of numbers occupying strategic squares. The only time the dead squares are involved with a winning-bingo combination is when the-bingo is made the "hard" way, 5 straight vertical numbers, or five straight horizontal numbers. All number selections for the regular and most of the special games require the use of only the strategic squares.

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