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Science column by Professor Auzin
Marcis Auzins: “Why read my texts? It seems to me that we often tend to “ignore” science, saying that it is formal, dry and uninteresting. I want the reader to see that they are a part of our lives – colorful and interesting.”
Biographical punctuation marks:
- Physicist by profession, currently professor at the University of Latvia, head of the Department of Experimental Physics and the Laser Center.
- From 2007 to 2015 he was rector of the University of Latvia.
- Works in the field of quantum physics and is the author of more than a hundred scientific articles published in the world’s leading physics journals, as well as several hundred conference proceedings.
- Together with colleagues from Riga and Berkeley, he wrote two monographs, which were published by Cambridge University Press and Oxford University Press and both were reprinted.
- Over the course of his career, he has lived and worked in various countries – China and Taiwan, the United States, Canada, England, Israel and Germany.
Let’s go back to Einstein’s quote. It took a very long time in human history for such a seemingly simple and intuitively clear concept of probability to be understood and accepted. There were several reasons for this. Perhaps the main thing is the feeling that when you roll the dice, the result is determined by fate or destiny, that is, by a higher will that decides the whole world. But attempting to describe fate or even the will of God mathematically is both heretical and arrogantly frivolous.
Dice games have been around for a very long time. Wild dice, almost the same ones used today, were found by archaeologists in excavations dating back several thousand years BC. 400 BC, both in ancient India and Mesopotamia and elsewhere in the world.
It is therefore even surprising that serious thinking about dice and probability in general as a matter capable of mathematical analysis did not begin until the 16th century. It was the Italian mathematician Gerolamo Cardano. Historians claim that Cardano himself was a big fan of the dice game and his interest in predicting its results came in very handy.
It turns out that once you roll the dice or flip a coin, it’s really impossible to predict what the outcome will be. But knowing the statistics when flipping a coin or rolling the dice multiple times is relatively safe. Mathematicians call this the law of large numbers. A large number in this case is the number of times a coin or dice is thrown. For example, if you flip a coin a thousand times, it’s safe to say that the number it lands with isn’t far from five hundred. The higher the number of shots, the more accurately the prediction is fulfilled.
I think that intuitively everything is relatively clear here and does not raise any questions. But does intuition always help?
Next example. We play a card game and are dealt four cards. The probability of getting exactly four aces, possibly the strongest cards in the game, is slim. To achieve this, you have to be very lucky. But now I’m wondering: What is the probability that there is a combination of cards that is nothing special, such as the Four of Clubs, the Ten of Clubs, the Servant of Spades and the Six of Clubs? For those who don’t play cards, I’ll explain that clubs, clubs, spades and clubs are four types of card suits in a deck of cards.
It may seem obvious to some, but not so obvious to others, that both situations – four aces and the aforementioned hand – are equally likely. The catch is that a queen’s ace seems to be very special, while a queen’s four seems unremarkable. But that’s just a subjective feeling – what importance we give to each card in our deck value scale. In fact, each card is just as special and unique as any other since there is only one in the deck. I hope I have convinced you.
Perhaps the next example will be even more unexpected for many. It is quite famous and is often mentioned. Let’s imagine a television play. The rules are simple. There are three locked doors. Behind two there is a goat and behind the third – a new, beautiful car. The player chooses one of the doors but does not open it. Then the game master opens one of the remaining two. But it is known for certain that if he knows what is behind each door, he will open the door behind which the goat is. Now it is clear that there is a car behind one and a goat behind the other. What’s the smart thing you can do as a player? Do you ask to open the door you originally chose, or do you change your choice and ask to open the other closed door? Intuition often says that there is no difference and the probability of getting a car is fifty to fifty percent no matter which door the player wants to open. But it is not. It turns out that by changing the original choice, the probability of getting a car is higher. It is possible to prove it mathematically, but I think it is not obvious to everyone. We leave the search for this proof to those readers who enjoy mathematics.
I would like to invite others to simply accept that not everything is as simple as it seems at first glance.
And finally, an example and an interesting question that shows that the answers are not so obvious. Let’s imagine there is a working group. The question is: How many people have to attend a party so that the probability that two of the party members have the same birthday increases by more than fifty percent?
It turns out the number isn’t that big. Twenty-three members are enough. How close was your estimate to this number?
But now a slightly different story. A party is not just a party, but you organized it because you are celebrating your birthday. How many people do you have to invite to your birthday so that there is over fifty percent chance that one of the other guests has a birthday that day? What is your guess?
Answer: More than two hundred and fifty guests should be invited to the party. My intuition would say the number will be lower, but the math isn’t wrong…
Such unexpected and sometimes seemingly paradoxical examples could be continued again and again.
By the way, casinos and gaming halls can no longer do without knowledge of the exact theory of probability these days. Every now and then there is the temptation to think of a game tactic in order to win with absolute certainty. But we can be absolutely sure that the calculation will work out and everything will be calculated and designed in such a way that only the arcade will always be in the black. This in turn means the sad truth that everyone else will ultimately be losers, even if they eventually succeed.
There is one tactic that might seem safe. In roulette, I place my bet – one euro – on the chance that an even number will come out. If I win, I get one euro. If I lose, I now bet two euros on an even number; If I win, I have one euro plus, because the winnings are four euros, but I put three into play in the previous rounds. If I lose again, I double the bet again. It is clear that you cannot roll an odd number forever, and at some point I will win and be one euro ahead.
However, as far as we know, the casino has also protected itself against this by setting a maximum rate up to which the amount can be doubled. Therefore, there is no surefire tactic.
What is certain is that even if it works at some point, in the long run only the casino will be the winner. Everyone else will eventually be in the red if they play for a long time.
This time we talked a lot about mathematics in calculating probabilities in games and gambling, but today probability theory is a developed branch of mathematics that is very widely used, among other things, for accurately assessing various security risks.
Science column by Professor Auzin
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