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I both adore and fear mathematical paradoxes. In fact, they are not paradoxical in terms of mathematics, because clearly obey its laws. But, the attempt of understanding them using the usual way of thinking causes dizziness. There is a feeling that you are in another world, with other rules.


I think many of you have heard about the <a href="https://en.wikipedia.org/wiki/Monty_Hall_problem">Monty Hall problem</a>. I'm not going to come up with new wording, so will take the quote from wiki:

> Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
<center> ![3.jpg](https://steemitimages.com/DQmTZBSgXwBbcftyDHmUfh9dtHiimopK6j9r1A71z4MmLgT/3.jpg)</center>

Initially, in the presence of 3 doors, the chance of winning the car was 1/3 (33.3%). After the actions of the host, the player has only two doors for choice. Our intuitive thinking shows that the chance of winning the car is 1/2 (50%). 

**But! If you do the math, it turns out: if the player will change the decision and choose another door (relative to original), his chances of winning the car increases to 2/3 (66.6%).**

**Don't worry!** In this post I will not analyze this paradox in details. It was done hundreds of times before me, and the number of digits causes a headache. I would like to tell you about a simple mathematical trick, which will allow you to understand the principle of all these paradoxes.

# <center>Hello, my name is Boris...</center>
<p>

**Imagine: You met a man (his name is Boris, for example). Boris says that there is two children in his family - HE and someone else. Question: what is the probability that Boris has a brother, not a sister?**
<p>

(*This is a standard mathematical task, which occurs in all textbooks on the theory of probability*)

Even knowing that this is a trick question and the answer is not obvious, your brain is still given the answer 50%/50% in automatic mode. Because we used to think like that. 

Really, what else could be the probability if there are only two sexes - male and female ?! Boris’s relative is either a man or a woman, only two objects to choose.

In the Kingdom of mathematics you could lose your head for this answer. So, just in case, let's see how *"their"* laws work.

# <center>Сombinations</center>
<p>

There are only 4 mathematical combination for two objects. A combination also implies the order in which the objects follow each other. In the case of birth of children in the family it looks like this:

<center> ![1.png](https://steemitimages.com/DQmVRrvAPXSH1MuQ4huDLcmMG2JPyu17zVASva4ihX7fAsB/1.png)</center>

* Male | Male     
* Male | Female 
* Female | Male 
* Woman | Woman


On the example of figures it looks even clearer. Two objects: 1 and 2. All possible combinations: 
* 1|1
* 1|2
* 2|1
* 2|2

We know that Boris is male. I.e. we can excluded combination **Woman| Woman** from the calculations. As a result, we have the following options: 

<center> ![2.png](https://steemitimages.com/DQme2TpytvxjKsRY7CBHPQq9jNgtqnQbtqm6TQFS8eHENhN/2.png)</center>

* Male | Male     
* Male | Female 
* Female | Male 

#### The chances that Boris has a sister is 2/3 (66.6%), against the probability of the existence of brother 1/3 (33.3%). In the world of mathematics this approach to computing is called <a href="https://en.wikipedia.org/wiki/Combinatorics">**combinatorics**</a>. In calculating the probability you take into account the influence of objects on each other.
<p>

The obtained numbers don't mean that in our world there is 2 times more women, than men. They only increase your chances to calculate the right answer to the question, or rather to find a more likely answer to the question.

**The secret of the trick (and others) is that our everyday thinking dominates the abstract. You would have immediately found the correct answer if saw the possible options directly in front of you.**

<center> ![393206026363282.jpg](https://steemitimages.com/DQmX9zB1Fh6XAYorepHREh1SKit23GJMJwddbDWwJYYssX5/393206026363282.jpg)</center>

#### Over the past century, people have developed unique abilities for abstract thinking. Due to this, we are able to look into distant galaxies and marching to meet the flying cars and biological immortality. Mathematics, like the science and culture of computing has gone far ahead of our innate biological ability to calculations, which primarily rests on the visible objects. Therefore, our everyday logic fails when confronted with similar examples.

****
*Read about other interesting mathematical paradoxes, some of them will allow you to play on a bet:*
- <a href="https://steemit.com/mathematics/@natord/penney-s-game-how-to-win-at-head-and-tails">Penney's Game: How to Win at "Head or Tails"</a>
- <a href="https://steemit.com/mathematics/@natord/mathematics-how-you-can-create-something-that-has-never-existed">Mathematics: How You Can Create Something That Has Never Existed Before</a>
- <a href="https://steemit.com/science/@natord/the-birthday-paradox">The Birthday Paradox</a>
****

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