December 29, 2014 at 06:58This is correct but not true.
Experts deservedly do not like tasks and puzzles at interviews. But we just love to solve such problems for our pleasure. This is what I personally don’t like when you get the right answer, but your decision seems to be wrong to the author. I just want to show the solution to several popular similar problems that can be obtained in the mind without complex calculations and compare them with the author's faithful ones.
Task 1In a country where people want only boy children, each family continues to give birth until a boy is born. If they have a baby girl, they have another baby. If a boy, they stop. What is the ratio of boys and girls in such a country? (It is understood that the probability of giving birth to a boy is equal to the probability of giving birth to a girl, although, in fact, the ratio is 105: 100) The
correct solution to this problem.
With a probability of 1/2 there will be one child - a boy.
With a probability of 1/4 there will be two children - a girl and a boy.
With a probability of 1/8 there will be three children - two girls and a boy.
...
With a probability of 1/2 ^ n there will be n children - (n-1) girls and a boy.
The mathematical expectation of the number of boys = 1
The mathematical expectation of the number of girls = 1/2 * 0 + 1/4 * 1 + 1/8 * 2 + 1/16 * 3 + ...
If the sum of this series is denoted by s, then it is easy to obtain that 2 * s - s = 1 , s is 1, therefore, the ratio of boys and girls is 50:50.
And now the correct solution to this problem.
All schemes in the condition of this task are for the removal of eyes. There is a Great Random that gives a boy and a girl with the same probability, which means the ratio will be 50:50.
If not convincingly, I will explain. Someone is playing martingale in a casino, that is, betting on a color until he drops out, doubling the bets. From this behavior of the player (or players), roulette will still produce the same number of black and red outcomes, exactly like the hospital.
Task 2Two firms sold coffee at the same price and volume. Both simultaneously held shares: the first started selling 15% more coffee, and the second 15% cheaper. Who has become more profitable to buy coffee from?
The right solution to this problem.
If before the action 100 ml of coffee cost 1 ruble, then after the action the first company 100 ml of coffee began to cost 100 * (100/115), which is approximately 87 cents, and the second, obviously, 85 or two cents cheaper.
And now the correct solution to this problem.
Divide by 115 in my mind, for example, is difficult for me. Therefore, it is possible to replace 15 with 50. In this case, the second company will receive twice as much coffee per ruble, and the first one only one and a half. The transition from 15 to 50 is legal in view of the linearity of interest.
Problem 2 bis.What is more profitable: a contribution at 70% in a currency with inflation of 60%, or at 80% in a currency with inflation of 70%?
Task 3. Here are two identical wine glasses. In one of them is wine, in the other is water. Scoop a teaspoon of water and pour into a glass of wine. Stir well. And then scoop a teaspoon of the mixture and pour into a glass with water. What is more: wine in a glass with water or water in a glass with wine?
The right solution to this problem.
1. Assume that there are 100 parts of liquid in a glass, and 10 parts in a spoon
2. Take 10 parts of water from a glass and pour into a glass with wine and mix
3. In a wine glass with wine 110 parts of a liquid. And in a spoonful of a mixture of this wine glass, one eleventh part of the volume of water and wine. Therefore, a spoonful of the mixture contains 9 whole and 1/11 parts of wine and 10/11 parts of water. All this is poured into a glass with water
4. Now in a glass with water there are 90 whole and 10/11 parts of water and 9 whole and 1/11 parts of wine, which in total gives 100 parts of liquid
5. In a glass with wine, 90 whole and 10 / 11 parts of wine and 9 whole and 1/11 parts of water, which in total also makes up 100 parts of liquid
6. Equivalent exchange
And now the correct solution to this problem.
Do not count the parts and how to stir. No matter what the manipulations take place, the entire withdrawn volume is replaced by the same arrived. And that’s all.
BonusTasks that are quickly and correctly solved. I present you the pleasure of solving them yourself.
1. Last Friday, the girl Masha went to the club for the first time and met 20 new people. This Friday she went to a club, met 10 old friends and met 10 new people. How many new people will the girl Masha most likely meet next Friday?
2. A convex polyhedron casts a pentagonal shadow. What is the minimum number of faces he has?
3. A triangle that does not have obtuse angles is called acute-angled. How to place twelve points in three-dimensional space so that they are the vertices of the largest number of acute-angled triangles? How many acute-angled triangles do you get?
4. You and another stranger are invited to make a natural number. If your numbers match, then you get a prize. What number do you guess?
5. In the box with cookies is an insert. To win, you need to collect a complete collection of different inserts, for this the average consumer buys 72 boxes. How many different inserts are there in the complete collection?
6. What thickness should the coin be (in radii) so that the probability of falling on an edge is equal to the probability of falling an eagle?
Task 1In a country where people want only boy children, each family continues to give birth until a boy is born. If they have a baby girl, they have another baby. If a boy, they stop. What is the ratio of boys and girls in such a country? (It is understood that the probability of giving birth to a boy is equal to the probability of giving birth to a girl, although, in fact, the ratio is 105: 100) The
correct solution to this problem.
With a probability of 1/2 there will be one child - a boy.
With a probability of 1/4 there will be two children - a girl and a boy.
With a probability of 1/8 there will be three children - two girls and a boy.
...
With a probability of 1/2 ^ n there will be n children - (n-1) girls and a boy.
The mathematical expectation of the number of boys = 1
The mathematical expectation of the number of girls = 1/2 * 0 + 1/4 * 1 + 1/8 * 2 + 1/16 * 3 + ...
If the sum of this series is denoted by s, then it is easy to obtain that 2 * s - s = 1 , s is 1, therefore, the ratio of boys and girls is 50:50.
And now the correct solution to this problem.
All schemes in the condition of this task are for the removal of eyes. There is a Great Random that gives a boy and a girl with the same probability, which means the ratio will be 50:50.
If not convincingly, I will explain. Someone is playing martingale in a casino, that is, betting on a color until he drops out, doubling the bets. From this behavior of the player (or players), roulette will still produce the same number of black and red outcomes, exactly like the hospital.
Task 2Two firms sold coffee at the same price and volume. Both simultaneously held shares: the first started selling 15% more coffee, and the second 15% cheaper. Who has become more profitable to buy coffee from?
The right solution to this problem.
If before the action 100 ml of coffee cost 1 ruble, then after the action the first company 100 ml of coffee began to cost 100 * (100/115), which is approximately 87 cents, and the second, obviously, 85 or two cents cheaper.
And now the correct solution to this problem.
Divide by 115 in my mind, for example, is difficult for me. Therefore, it is possible to replace 15 with 50. In this case, the second company will receive twice as much coffee per ruble, and the first one only one and a half. The transition from 15 to 50 is legal in view of the linearity of interest.
Problem 2 bis.What is more profitable: a contribution at 70% in a currency with inflation of 60%, or at 80% in a currency with inflation of 70%?
Task 3. Here are two identical wine glasses. In one of them is wine, in the other is water. Scoop a teaspoon of water and pour into a glass of wine. Stir well. And then scoop a teaspoon of the mixture and pour into a glass with water. What is more: wine in a glass with water or water in a glass with wine?
The right solution to this problem.
1. Assume that there are 100 parts of liquid in a glass, and 10 parts in a spoon
2. Take 10 parts of water from a glass and pour into a glass with wine and mix
3. In a wine glass with wine 110 parts of a liquid. And in a spoonful of a mixture of this wine glass, one eleventh part of the volume of water and wine. Therefore, a spoonful of the mixture contains 9 whole and 1/11 parts of wine and 10/11 parts of water. All this is poured into a glass with water
4. Now in a glass with water there are 90 whole and 10/11 parts of water and 9 whole and 1/11 parts of wine, which in total gives 100 parts of liquid
5. In a glass with wine, 90 whole and 10 / 11 parts of wine and 9 whole and 1/11 parts of water, which in total also makes up 100 parts of liquid
6. Equivalent exchange
And now the correct solution to this problem.
Do not count the parts and how to stir. No matter what the manipulations take place, the entire withdrawn volume is replaced by the same arrived. And that’s all.
BonusTasks that are quickly and correctly solved. I present you the pleasure of solving them yourself.
1. Last Friday, the girl Masha went to the club for the first time and met 20 new people. This Friday she went to a club, met 10 old friends and met 10 new people. How many new people will the girl Masha most likely meet next Friday?
2. A convex polyhedron casts a pentagonal shadow. What is the minimum number of faces he has?
3. A triangle that does not have obtuse angles is called acute-angled. How to place twelve points in three-dimensional space so that they are the vertices of the largest number of acute-angled triangles? How many acute-angled triangles do you get?
4. You and another stranger are invited to make a natural number. If your numbers match, then you get a prize. What number do you guess?
5. In the box with cookies is an insert. To win, you need to collect a complete collection of different inserts, for this the average consumer buys 72 boxes. How many different inserts are there in the complete collection?
6. What thickness should the coin be (in radii) so that the probability of falling on an edge is equal to the probability of falling an eagle?