In this week, we're going to talk about some of the social and ethical issues surrounding gambling. Before I start, it's important to stress that gambling is for many people a controversial subjects and many people are deeply opposed to it. It's certainly true to say that gambling can cause problems, and we don't want to suggest that as a result of studying these MOOCS, you should go out and start gambling. I think there are two issues to bear in mind when thinking about the issue of gambling in practice. The first is that, it is very unlikely that you can systematically make money from gambling. There are people who claim that they do this, and it's certainly true that you can get lucky, but you can also have bad luck. The evidence seems to suggest that, on average, people who gamble regularly can do little better than break even. Indeed in many cases will in the long-term lose money. Maybe not necessarily that much money, but it's not really a long term profitable investment. The only reason to engage in gambling should really be for fun, and you should think of the money you spend on gambling is being part of having that fun. If you start to lose large sums of money, the fun will go out of it, and we don't want to encourage anybody to do that. The second thing to say I think is that, gambling can ultimately for some individuals, become a real problem, and anybody who thinks that they might be getting into trouble should always seek out help. Gambling addiction can be very harmful not only to the individual, but to the people around them. Again, we don't want to encourage excessive gambling. If you do it, it should be for fun and preferably for really small amounts of money that you can afford to lose rather than betting the house on it. The development of probability theory was not surprisingly came from mathematicians who worked on problems associated with probability. But what's less familiar is that, many of those problems that the early mathematicians were working on were actually gambling problems, and we're going to just take a review of some of the major figures in the evolution of probability theory as it relates to gambling. Gambling itself is as old as the hills, so we know that the ancient Greeks and the Romans involved were engaged in gambling, and gambling in China goes back at least 4000 years. We know that it's an activity that humans have engaged in. However, what's interesting is perhaps that the sense of the notion of a probability associated with gambling is something that doesn't seem to be merged until much later. It really seems to be only in the 16th century in Europe that mathematicians started to think systematically about probabilities and their relationship to gambling. Before that, it seems that people thought that the outcome of games of chance was determined by fate, which was something that was fixed and unknowable, so that you didn't have a concept that some things were more likely than others. But of course, once mathematician start to look at gambling problems, they start to realize, for example, if you think about throwing two dice, then you know that the probability that the sum of numbers on a six-sided dice will add up to, if you throw twice, then the numbers will add up to seven is much more likely that the numbers will add up to 12 or two. In that sense, as people started to think about what were likely outcomes, these started to raise the thought of probability as something that could be studied and understood. Perhaps the first great figure in the evolution of the theory of probability is Gerolamo Cardano, who worked in Italy in the 16th century. Now, there may have been people who wrote something which looked a little bit like the concept of probability, and suddenly they were precursors even before him. But his work specifically formalized the idea that they were for each problem, there were a distinct number of events that could occur, and each would have a probability. These probabilities should ultimately add up to one. Once you take into account all the possibilities that defines the total universe of possible outcomes, and that stem from following gambling problems. In fact, he's wrote his book, which in Latin means a book on games of chance, shows just how closely tied he was to gambling problems. Two of the greatest mathematicians in history are Pierre Fermat and Blaise Pascal. They were in correspondence in the 17th century. They both studied a particular problem that stemmed from gambling. This problem is a classic practical issue in gambling. The idea here is that you play a game in a number of rounds. Here the example is 6 rounds, and the winner of the 6 rounds wins a prize. Then a problem that would arise in these games is that the game would unexpectedly had to stop, perhaps because gambling was illegal and the police were coming around, so you needed to stop the game. If you stop the game after a fixed number of rounds, how should the price be divided between A and B depending on the success up to that point in the game. This is known as the "Problem of Points." Both Fermat and Pascal provided their own solutions to this. Pascal's triangle, which most people have learned at school at some point that was basically heed the framework that he developed to opt for answering that problem. One of the greatest figures in the early development of probability theory is Christian Huygens who was a multi-talented individual inventor, physicist, astronomer. He invented the pendulum clock, which was a fundamental step forward in the measurement of time. He is credited with developing the concept of an expected value. The idea of an expected value is that we can identify in a probabilistic way what the likely outcome is going to be based on the average of outcomes from a series of experiments. This concept of expected value is fundamentally significant in statistics and something which is stand behind when we calculated the mean value for results in looking at, say, premier league games, we will really be using that in the sense of an expected value. He formulated problems like the "Gambler's Ruin," a problem which is an interesting problem to think through for yourself and see if you can come up with a solution. Abraham de Moivre was a Frenchman who lived most of his life in Great Britain. He was one of the great early probability theorists as well. He came up with a similar problem. A lot of these problems had something in common. Solving a problem where you have two players, each with a fixed number of counters. Each time you lose around, you pay one counter to the winner. The loser pays the winner. What's the probability that you win all of the counters within a specified number of rounds? In essence, all of these problems really come down to counting. Probability theory is in many ways just a problem of counting. Just, what are the possibilities and how do we enumerate all of those possibilities and turn those possibilities into probabilities. Jacob Bernoulli, in the end of the 17th century, beginning of the 18th century, was a member of a family that produced some of the greatest mathematicians. He is credited with proposing the law of large numbers. Everyone who says that the likely outcome of a coin toss is 50 percent heads, 50 percent tails as long as it's a fair coin, is really probably has in mind the Jacob Bernoulli's results, which essentially stated this as based on the idea that as long as you have enough coin tosses, it's almost inevitable that the number of heads will be 50 percent, and that if the law of large numbers works, then eventually you do get to 50 percent at some point. He formulated some other interesting problems which he wrote down in his book, The Art of Conjecture, which developed solutions to gambling problems such as those proposed by Huygens. Jacob's nephew Daniel Bernoulli, is famous for solving a famous paradox in probability, the St. Petersburg Paradox. The St. Petersburg Paradox emerges from a gambling proposition. Suppose we have a gamble where you toss a coin repeatedly. The way this works is you call heads or tails, and if it comes down heads the first time you win two dollars or if it comes down tails, you lose and the game ends. Then if you win, you go on to a second round. In the second round, you toss a coin and if it comes down heads, again, you win double the amount, you win four dollars or if you lose, the game ends. You then go to a third round, where again, you toss a coin and if it comes down heads, you double the value, so you get eight dollars, but if it comes down tails, you lose and you're out. You keep playing this game doubling the pay if you win each round, ending the game if you don't. How much would you pay to play this game? This puzzle people, because on the face of it, you think you'd pay an infinite amount, because the first time you've got a 50 percent chance of winning two dollars, which has a value of a dollar in expectation. In the second time you have a 25 percent chance of winning four dollars, so then you'd be willing to pay a dollar for that. The third round you get a 1/8 chance of getting eight dollars and so that should also be worth a dollar. Every time, every round should have a value of a dollar. If you keep on playing this game forever, you should have an infinite value, but no one would pay an infinite amount of money to play this game. Daniel then proposed the concept of risk aversion to explain this phenomenon, which is a concept that is still widely used in statistics or economics and social science in general, this idea that the amount that we're prepared to pay for a fair gamble may be much less than its expected value, largely because we are averse to losing large sums and would rather have a smaller sum with certainty than a larger sum with some risk that we lose significantly. The line of mathematicians who developed probability theory around gambling really comes to an end with the Pierre-Simon Laplace, who was working in the 18th and beginning of the 19th century. He generalized a lot of these problems that the mathematicians have been working on for many years, and he advanced mathematics in many other ways. By the time of Laplace, the interest in gambling as a way of understanding probability was diminishing and that had to do with really the fact that gambling problems no longer provided significant challenges. The mathematicians felt they'd solve most of the problems and the challenges were actually moving into other areas and more complex problems, say to do with insurance and working out, for example, how much should an insurance company charge for life insurance, given the probability that you will die within some fixed period. Thinking about investment problems, what's your expected return on investment in the stock market and so on. The mathematicians' focus shifted away from some of the pure gambling problems and thinking about more general problems. Although, as I observed during an earlier session, there is a congruence between investment in the stock market and gambling problems in general. Probability theory is involved in pretty much every aspect of research in natural sciences and social sciences, even the fundamental theories of the universe which associate with quantum mechanics come down to probability theory. In some sense, all of it comes down to gambling. It's ironic then to think of Einstein's famous comment about quantum mechanics which horrified him, since it's based on probability theory. He said, God does not play dice with the universe. Well, maybe that's true, maybe that's not true. But certainly dice players have had a lot to do with the development of probability theory throughout history. We have at least that to thank gambling for. The development of probability theory has been closely related to solving statistical problems involved in gambling. Some of the greatest mathematicians in history devoted themselves to these subjects. Still today in the academy you will still find many professors will introduce basic explanations of probability theory through the use of very simple gambling problems. That doesn't mean to say you should gamble, but it does mean to say that gambling has had its uses.