On a Study of some Gambling Games

Introduction

Probability theory began in seventeenth century France when the two great French mathematicians, Blaise Pascal and Pierre de Fermat, corresponded over two problems from games of chance. Problems like those Pascal and Fermat solved continued to influence such early researchers as Huygens, Bernoulli, and DeMoivre in establishing a mathematical theory of probability. Today, probability theory is a well established branch of mathematics that ends applications in every area of scholarly activity from music to physics, and in daily experience from weather prediction to predicting the risks of new medical treatments.

Before the theory of probability was formed Gambling was popular. Gamblers were crafty enough to figure simple laws of probability by witnessing the events at first hand. The opportunity was limitless in then exploiting the often complex and sometimes seemingly contradictory laws of probability.

In the seventeenth century Galileo wrote down some ideas about dice games. This led to discussions and papers which formed the earlier parts of probability theory. There were and have been a variety of contributors to probability theory since then but it is still one of the least understood areas of mathematics.

De Mere's paradox

Which is more likely, getting at least one six in 4 rolls of a die (Game I), or getting at least one double six in 24 rolls of 2 dice (Game II)? The Chevalier de Mere (1610--1685) argued (incorrectly, as it turned out) that both probabilities are the same (and equal to 2/3), since 4 times 1/6 is 2/3 and 24 times 1/36 is also 2/3. What are the correct probabilities for winning in these two games? Which (if any) of the two games gives you a better than 50-50 chance of winning?

We often hear that the theory of probability started in the seventeenth century, when a French nobleman, the Chevalier de Méré, proposed the following problem in 1654 to his friend Pascal: Why is one more likely to obtain a “6” in four throws of a die than to obtain a double “6” in 24 throws of two dice? This problem is known as de Méré’s paradox. We use the word paradox, because, based on the fact that there are 6 possible results when we roll a die and 36 possible results when we roll two dice, some people thought that the two events above should have the same probability. Indeed, notice that the number of throws, divided by the number of possible results, is equal to 2/3 in both cases (4/6 = 24/36 = 2/3). Nowadays, we can easily compute the probability of each event. We find that the probability of obtaining at least one “6” in four rolls of a (fair or non-biased) die is 1 − (5/6)4 = 671/1296 ≃ 0.5177, while the probability of getting at least a double “6” in throwing two dice 24 times is 1 − (35/36)24 ≃ 0.4914.. We can deduce that the Chevalier de Méré must have spent a lot of time throwing dice to discover such a small difference!

Simulations

It is said that de Mere had been betting that, in four rolls of a die, at least one six would turn up. He was winning consistently and, to get more people to play, he changed the game to bet that, in 24 rolls of two dice, a pair of sixes would turn up. It is claimed that de Mere lost with 24 and felt that 25 rolls were necessary to make the game favorable. It was un grand scandal that mathematics was wrong.

We shall try to see if de Mere is correct by simulating his various bets. There have been developed a program DeMere1 that simulates a large number of experiments, seeing, in each one, if a six turns up in four rolls of a die. When we ran this program for 1000 plays, a six came up in the first four rolls 48.6 percent of the time. When we ran it for 10,000 plays this happened 51.98 percent of the time.

We note that the result of the second run suggests that de Mere was correct in believing that his bet with one die was favorable; however, if we had based our conclusion on the first run, we would have decided that he was wrong. Accurate results by simulation require a large number of experiments. 2

The program DeMere2 simulates de Mere’s second bet that a pair of sixes will occur in n rolls of a pair of dice. The previous simulation shows that it is important to know how many trials we should simulate in order to expect a certain degree of accuracy in our approximation. We shall see later that in these types of experiments, a rough rule of thumb is that, at least 95% of the time, the error does not exceed the reciprocal of the square root of the number of trials. Fortunately, for this dice game, it will be easy to compute the exact probabilities. We shall show in the next section that for the first bet the probability that de Mere wins is 1 − (5/6)4 = .518. One can understand this calculation as follows: The probability that no 6 turns up on the first toss is (5/6). The probability that no 6 turns up on either of the first two tosses is (5/6)2. Reasoning in the same way, the probability that no 6 turns up on any of the first four tosses is (5/6)4. Thus, the probability of at least

one 6 in the first four tosses is 1 − (5/6)4. Similarly, for the second bet, with 24 rolls, the probability that de Mere wins is 1 − (35/36)24 = .491, and for 25 rolls it is 1 − (35/36)25 = .506.

Using the rule of thumb mentio