Professor Auzin’s science column
Marcis Auzins: “Why read my texts? It seems to me that we often tend to “ignore” natural sciences, saying that they are formal, dry and uninteresting. I would like to let the reader see that they are a part of our lives – colorful and interesting.”
Biography punctuation marks:
- A physicist by profession, currently a professor at the University of Latvia, head of the Department of Experimental Physics and the Laser Center.
- From 2007 to 2015, he was the 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 and several hundred conference reports.
- Together with colleagues from Riga and Berkeley, he wrote two monographs, published by “Cambridge University Press” and “Oxford University Press”, both of which have been reprinted.
- During his career, he lived and worked in different 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 will be determined by fate or providence, that is, some higher will that decides the whole world. But trying to mathematically describe fate or even God’s will is both heretical and arrogantly frivolous.
Dice games have been known for a very long time. Game dice, almost the same as those used today, have been found by archaeologists in excavations dating back several thousand years BC, both in Ancient India and Mesopotamia and elsewhere in the world.
Therefore, it is even unexpected that serious thinking about the game of dice and probability in general as a mathematically analyzable matter began only in the 16th century. It was the Italian mathematician Gerolamo Cardano. Historians claim that Cardano himself was a great fan of the dice game and his interest in predicting its results was very practical.
It turns out that it’s really impossible to predict what the outcome will be by tossing a dice or tossing a coin once, but knowing what the statistics will be by tossing a coin or die many 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 die is tossed. For example, if you toss a coin a thousand times, it is safe to say that the number of times a tail will land will not be far from five hundred. The higher the number of shots, the more accurately the prediction is fulfilled.
I think that intuitively everything here is relatively clear and does not raise questions. But does intuition always help?
Next example. We are playing a card game and we are dealt four cards. The probability of exactly four aces, possibly the strongest cards in the game, is slim. For that to happen, you have to be very lucky. But now I will ask, what is the probability that there will be 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 will explain that clubs, clubs, spades and clubs are four types of card suits in a deck of cards.
It may seem obvious to some, not so obvious to others, that both situations – four aces and the mentioned hand – are equally likely. The catch lies in the fact that the ace of a queen seems to be something very special, but the four of a queen is unremarkable. But this is only a subjective feeling – what importance we assign to each card in our card game value scale. In fact, each card is as special and unique as any other, as there is only one in the deck. I hope I convinced you.
Perhaps the next example will be even more unexpected for many. It is quite famous and often mentioned. Let’s imagine a television game. The rules are simple. There are three closed doors. Behind two 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 leader opens one of the remaining two. But it is known for sure that he, knowing what is behind each door, will open the door behind which the goat is. Now it is clear that of the remaining doors, behind one is a car, behind the other is a goat. What is the smart thing to do for a game player? Ask to open the originally chosen door or change your choice and ask to open the other closed door? Often, intuition says that there is no difference and the probability of getting a car is fifty percent to fifty, no matter which door the player asks to open. But it is not. It turns out that by changing the initial 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 will leave the search for this proof to those readers who enjoy mathematics.
I will invite the others to simply accept that not everything is as simple as it seems at first glance.
And finally, another example and an interesting question that shows that the answers are not so obvious. Let’s imagine that there is a work party. The question is this – how many people must attend a party in order to have more than a fifty percent probability that two of the party members will have the same birthday?
It turns out that the number is not so great. Twenty-three members are enough. How close was your guess to that number?
But now a slightly different story. A party is not just a party, but you have organized it because you are celebrating your birthday. How many people do you need to invite to your birthday party so that there is a greater than fifty percent chance that one of the other guests has a birthday on that very day? What’s your guess?
Answer – more than two hundred and fifty guests should be invited to the party. My intuition would say that the number will be less, but the math is not wrong…
Such unexpected and sometimes seemingly paradoxical examples could be continued again and again.
By the way, nowadays casinos and gaming halls cannot do without knowledge of the exact theory of probabilities. Every now and then there is a temptation to come up with a game tactic in order to win with absolute certainty. But we can be absolutely sure that the math works and everything is calculated and created so that the gaming hall is the only one that will always be in the pluses. This in turn means the sad truth that everyone else will end up being a loser, even if they can succeed at some point.
There is one tactic that might seem safe. In the roulette game, I place my bet – one euro on the fact that an even number will fall out. If I win, I get one euro. If I lose, now I bet two euros on an even number; if I win, I have one euro plus, because the winnings are four euros, but I have put three in the game in the previous rounds. If I lose again, I double the bet again. It’s clear that you can’t keep rolling an odd number forever, and I will eventually win at some point and be one euro ahead.
However, as far as we know, the casino has also secured against this by setting a maximum rate up to which the amount can be doubled. Thus, there is no surefire tactic.
What is certain is that, even though it may work out at some point, the winner in the long game will only be the casino. Everyone else will eventually be in the red when playing long.
This time we talked a lot about mathematics in calculating probabilities in games and gambling, but nowadays probability theory is a developed branch of mathematics that is used very widely, including for the accurate assessment of various security risks.
Professor Auzin’s science column
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2024-01-26 05:31:47
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