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Bad. Just terrible. Right?
Well, first of all let's make sure we are on the same page when it comes to the definitions. According to the dictionary, the word gamble means "play games of chance for money" or in a more general context "take risky action in the hope of a desired result.". An obvious example is lotto where you can place some money on some six distinct numbers from a range of 1-49 and if the six numbers on a ticket match the numbers drawn by the lottery, you win a (usually) crazy amount of money. The reason why this game is terrible is that the chance of winning in this game is 1 in 13,983,816. I.e almost zero. Similar arguments apply to most of the games where you bet money. So there is no logical reason to take part in those. Besides, most of the companies running them are evil and completely dishonest. By the way, even though the chances of winning the lottery are exceedingly low, it doesn't stop people from playing it: U.S. adults spend approximately 310 US dollars per person annually on lottery tickets, on average. It is 102 billion dollars a year spent on lottery per that country!
But it is by far not only lotto or similar games that fit that definition. In some way, almost any purchase and most of the actions fall under the category of gambling. Think about it, the second definition from above is merely about taking risky actions — and all kinds of activities can be called "risky"...
Exercise 1.
A bus ticket costs 3 pounds, and the penalty fare for not having a ticket is 40 pounds. If $p$ is the probability you meet a ticket inspector, for which $p$ it is more profitable (in expectation) not to buy a ticket?
Explanation and comments
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There is a formal way of doing this problem, but let's do it informally as it will be more intuitive then. So what does say $p=\frac{m}{n}$ mean? It means that out of $n$ trips, we expect the inspector to come $m$ times. So let's compare two strategies on $n$ trips:
This could be applicable to life if only we knew the real $p$... And well, if we did not care that much about the moral side of actions.
Let's get a bit more serious. If you think about it, any investment is basically gambling. In fact, understanding risk in a financial setting is precisely what mathematical finance is concerned about. It is okay if it is the first time you hear the words "mathematical finance"; what is important here is that it is something that has to do with mathematics and has a special name (it is an entire field of mathematics!). Moreover, tens of thousands of people do it while consistently earning a lot of money. The most famous of them is Warren Buffet, who has a net worth of over 100 billion US dollars as of November 2022 and who was literally the richest person on the planet in 2008.
It is not easy to become a strong mathematician-financier. Moreover, far not everyone may find it appealing: those people do not create anything amazing or useful for the world as a part of their job. Of course, if one becomes a good risk manager and donates a lot of his/her earnings to good charities, it will have a positive impact. But it is not a part of the job, and some finance people reinvest in evil yet profitable organisations. Plus, it can be stressful, and the working hours in some companies are insane. Therefore please do not see this text (or the course) as a call to become a trader or an investment banker.
Long story short, all we want you to take from this text are the following two things: first, you know about this career path, and second, you have at least some idea of what it is about. To progress with the latter, let's do an exercise below:
Exercise 2.
There is a plot of how the price of something was evolving. Can you spot a potentially-useful correlation here? It should help predicting the future movement of the price (up or down).
Explanation and comments
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The key observation here is that almost every the price goes up, it goes up again in the next hour and if the price drops, it drops in the next hour as well. This is not a totally "random" behaviour if you think about it. If it were a totally random behaviour, then right after the growth, we would expect a downgrade of around 50% of the time. I.e. the next price change is positively correlated with the last price change.
This phenomena is called positive autocorrelation (guess what kind of behaviour is called negative autocorrelation), a fancy, but self-explanatory, term. You could also call it a "trend".
Exercise 3.
By making use of the positive autocorrelation phenomena, find a way (=algorithm) to almost never have less than 1050 \$ dollars in the "total worth" by the end of the trading game below. Ideally also try to sometimes get more than 1150 \$ by the end. It should be clear what is going on, but here are the details just in case:
Explanation and comments
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This exercise is merely about getting your head around what that positive autocorrelation phenomenon means. It sounds fancy and scary, but it is just the idea of "trend".
Once again, intuitively, this phenomenon means that if last time the price went up, it means that the price is more likely to go up again. On the other hand, if the price just dropped, it is more likely to drop again. With this in mind, let's stick to the following strategy:
This is not a $100 \%$ winning strategy, but we are doing probability theory, i.e theory of chance :) We can further modify this strategy by the way: we can do "Sell All" and then always "Do Nothing" once our total worth is above $1050 \$ $. This will be an almost-always (~90% of the time) working strategy to finish with $>1050 \$ $ at the end. It will not, however, ensure that we get to $> 1200 \$$ sometimes. For this, we do need to stick to the strategy above until the end, at least every once in a while — but then we increase the risk of finishing with less than $1050 \$ $. Thus, there is some trade-off between risk and expected final total worth.
This was the last lesson of this introductory and exploratory course, and there are no further problem sets. We hope that you are more convinced that "probability is everywhere" and that you have a better idea of what kind of maths and intuition will play a role in the applications.
This course has been designed to be beginner-friendly and short. Therefore, a few spheres where probabilistic theories are important were barely even mentioned. E.g Statistical Physics, Medicine or Cryptography. Once we talk more about maths and formalism, we will get to those as well, don't worry. However, if you are indeed convinced that knowing probability well is cool and super useful, then that should not matter to you as much: you can now focus a bit more on mastering the applicable and worldwide desired maths skills, practise applying them in all kinds of settings and while doing so (or maybe later) decide where exactly you want to use them.
Hopefully this journey has been useful for you, good luck!