Taking a situation to the extreme, in some cases, helps explain it better. A very good example is the Monty Hall problem. I came across it when the professor of my probability course gave it as an exercise. The solution of the problem is very surprising and I urge you to solve it for yourself before looking at the explanation below.
The problem is posed as a game played between you and a computer. Suppose that there are three identical rooms with doors that are controlled by the computer. Two of the rooms have a goat inside, but the third has a Porsche. You have no knowledge about the contents of any of the rooms but the computer has. The game has four steps (a) You pick a door, (b) The computer picks one of the two leftover doors and opens it; revealing a goat, (c) The computer gives you an option to switch to the other door, and (d) The computer opens the door you picked and you get the content of the room.
Step (b) needs some additional explanation. There are two situations. In Step (a) you might have picked the door with the Porsche behind it. In this case, the computer picks a door at random out of the other two i.e., picks one of the two with probability 1/2. In Step (a) if you pick a door with a goat behind it, then out of the two left-over doors, only one has a goat. The computer then opens this door to reveal a goat.
Rationally, you want to win the Porsche. The question is: in Step (c) should you switch the doors ?
Taking this situation to the extreme might help you make a decision. Instead of three doors, consider 100 doors with 99 goats and 1 porsche. In Step (b) computer opens 98 of the left-over 99 gates. Do you switch the doors in Step (c) ?
***
World War I was an expensive affair. The participants spent more and more money in the hope that they will be able to collect it from the group that lost the war. The German side lost and were demanded to pay nearly $64b. The GDP of Germany was $12b before the war and $64b was an extremely steep sum.
Germany had financed most of the war by printing money. It had expanded its money supply by 400% and the prices were nearly 10 times in 1920 compared to 1913. Instead of trying to rebuild its finances, the Germany took the easier way of inflating away the debt by printing as much money as needed. In fact, some claim that the Germans intentionally destroyed their economy to obviate the need of paying reparations.
The amount of inflation can be gathered from the fact that after the war $1 was worth 8.91 marks and on Nov 12, 1923 it was worth 630b marks. In Nov, just as the new currency Rentenmark was introduced, the mark fell some more. On Nov 14, it was at 1.3 trillion; on Nov 15, it was 2.5 trillion and on Nov 20, it was trading for 4.2 trillion !
In 1922, nearly 1 trillion marks were printed. In the first six months of 1923, the amount rose to 17 trillion (see Lords of Finance by Liaquat Ahamed, page 93-94).
***
Printing money, in simple terms, is a way to transfer wealth from the savers to borrowers. In the spirit of taking situations to the extremes, we can look at Germany and see what happened.
The savers of Germany (among them: teachers, civil servants, scientists, doctors, workers) were hit the hardest. They had invested all their life savings into government bonds and saving accounts. The careful accumulation of the money amounted to naught. A thousand mark bond from 1914 was still worth the same in 1923, but a loaf of bread cost in billions of marks. It is not difficult to see that the middle class Germany was reduced to indigence and penury.
The debtors (among them: the industrialists, real estate owners, risk takers with large debt) flourished and made like robbers. Their assets increased in price because of the inflation while their debt was worth nothing. A debt of 1000 mark in 1913 was still worth 1000 mark in 1923 (except that one has to pay the interest during the intervening 10 years). The debtors benefited from buying hard assets on margin.
***
Real-estate companies are excellent hedge against inflation. A real-estate company owns buildings and lands. This is a capital intensive business and generally the company has to take a lot of debt to finance the acquisition of new houses and new land.
Inflation will profit this company on two fronts (a) the prices of its assets increase, and (b) The debt will be worth a lot less.
For example, if you took 8.9 mark in loan and bought a piece of land in 1918 Germany then in 1923 the land will be worth a lot, while the loan will be still worth 8.9 marks !
***
Let us come back to the Monty Hall Problem. The solution to the problem is that switching at Step (b) is the correct choice. This way you double the probability of winning the car.
This answer is counterintuitive, even after some serious mental exercise. The car is either behind the door you chose or not. How can showing a goat in one of the remaining rooms change this fact ?
Taking the case to the extreme with 100 rooms explains why switching might help. The probability of picking the right door in Step (a) with the Porsche behind it, is 1/100. After the computer has shown 98 goats by opening 98 of the remaining 99 rooms, we have 2 left-over closed doors. If we pick one randomly among these two, the chance of hitting the Porsche is 1/2 (there are two gates and Porsche is behind one of them).
Let us answer the original problem more formally.
There are three ways of hiding the Porsche among the three rooms. If you pick the door with the Porsche in Step (a), switching in Step (c) makes you win a goat. If you pick the door with a goat in Step (a), then switching in Step (c) makes you win a Porsche. The probability that you picked the room with Porsche in it - in Step (a) is 1/3. The probability of picking a door with a goat behind it is 2/3 in Step (a). So, switching the door in Step (c) will make you win with probability 2/3, while staying with the same door will make you win with probability 1/3.
The problem is posed as a game played between you and a computer. Suppose that there are three identical rooms with doors that are controlled by the computer. Two of the rooms have a goat inside, but the third has a Porsche. You have no knowledge about the contents of any of the rooms but the computer has. The game has four steps (a) You pick a door, (b) The computer picks one of the two leftover doors and opens it; revealing a goat, (c) The computer gives you an option to switch to the other door, and (d) The computer opens the door you picked and you get the content of the room.
Step (b) needs some additional explanation. There are two situations. In Step (a) you might have picked the door with the Porsche behind it. In this case, the computer picks a door at random out of the other two i.e., picks one of the two with probability 1/2. In Step (a) if you pick a door with a goat behind it, then out of the two left-over doors, only one has a goat. The computer then opens this door to reveal a goat.
Rationally, you want to win the Porsche. The question is: in Step (c) should you switch the doors ?
Taking this situation to the extreme might help you make a decision. Instead of three doors, consider 100 doors with 99 goats and 1 porsche. In Step (b) computer opens 98 of the left-over 99 gates. Do you switch the doors in Step (c) ?
***
World War I was an expensive affair. The participants spent more and more money in the hope that they will be able to collect it from the group that lost the war. The German side lost and were demanded to pay nearly $64b. The GDP of Germany was $12b before the war and $64b was an extremely steep sum.
Germany had financed most of the war by printing money. It had expanded its money supply by 400% and the prices were nearly 10 times in 1920 compared to 1913. Instead of trying to rebuild its finances, the Germany took the easier way of inflating away the debt by printing as much money as needed. In fact, some claim that the Germans intentionally destroyed their economy to obviate the need of paying reparations.
The amount of inflation can be gathered from the fact that after the war $1 was worth 8.91 marks and on Nov 12, 1923 it was worth 630b marks. In Nov, just as the new currency Rentenmark was introduced, the mark fell some more. On Nov 14, it was at 1.3 trillion; on Nov 15, it was 2.5 trillion and on Nov 20, it was trading for 4.2 trillion !
In 1922, nearly 1 trillion marks were printed. In the first six months of 1923, the amount rose to 17 trillion (see Lords of Finance by Liaquat Ahamed, page 93-94).
***
Printing money, in simple terms, is a way to transfer wealth from the savers to borrowers. In the spirit of taking situations to the extremes, we can look at Germany and see what happened.
The savers of Germany (among them: teachers, civil servants, scientists, doctors, workers) were hit the hardest. They had invested all their life savings into government bonds and saving accounts. The careful accumulation of the money amounted to naught. A thousand mark bond from 1914 was still worth the same in 1923, but a loaf of bread cost in billions of marks. It is not difficult to see that the middle class Germany was reduced to indigence and penury.
The debtors (among them: the industrialists, real estate owners, risk takers with large debt) flourished and made like robbers. Their assets increased in price because of the inflation while their debt was worth nothing. A debt of 1000 mark in 1913 was still worth 1000 mark in 1923 (except that one has to pay the interest during the intervening 10 years). The debtors benefited from buying hard assets on margin.
***
Real-estate companies are excellent hedge against inflation. A real-estate company owns buildings and lands. This is a capital intensive business and generally the company has to take a lot of debt to finance the acquisition of new houses and new land.
Inflation will profit this company on two fronts (a) the prices of its assets increase, and (b) The debt will be worth a lot less.
For example, if you took 8.9 mark in loan and bought a piece of land in 1918 Germany then in 1923 the land will be worth a lot, while the loan will be still worth 8.9 marks !
***
Let us come back to the Monty Hall Problem. The solution to the problem is that switching at Step (b) is the correct choice. This way you double the probability of winning the car.
This answer is counterintuitive, even after some serious mental exercise. The car is either behind the door you chose or not. How can showing a goat in one of the remaining rooms change this fact ?
Taking the case to the extreme with 100 rooms explains why switching might help. The probability of picking the right door in Step (a) with the Porsche behind it, is 1/100. After the computer has shown 98 goats by opening 98 of the remaining 99 rooms, we have 2 left-over closed doors. If we pick one randomly among these two, the chance of hitting the Porsche is 1/2 (there are two gates and Porsche is behind one of them).
Let us answer the original problem more formally.
There are three ways of hiding the Porsche among the three rooms. If you pick the door with the Porsche in Step (a), switching in Step (c) makes you win a goat. If you pick the door with a goat in Step (a), then switching in Step (c) makes you win a Porsche. The probability that you picked the room with Porsche in it - in Step (a) is 1/3. The probability of picking a door with a goat behind it is 2/3 in Step (a). So, switching the door in Step (c) will make you win with probability 2/3, while staying with the same door will make you win with probability 1/3.
