Some probability problems to think about.
1) Suppose you are on a game show. There are 3 doors, one of them revealing a car behind it, or 1 million dollars, or anything you like. The other two have nothing behind them. You can only pick a door. So you pick a door, let's say door A. The host, who knows which door has the grand prize and which doors do not, then open one of the other 2 doors, either B or C, which has nothing. He will always open a door which has nothing behind it. So the question is, should you stick with your original choice, or should you switch, or it does not really matter? What are the chances of you winning the grand prize if you do not switch? Could it be 1 in 3, because your original choice was a 1 in 3 chance so whatever the host does, does not make a difference, or could it be 1 in 2, because now you have 2 choices, either your original door or the door that is left?
2) Suppose you know this woman who has exactly two children, but you do not know their genders. You ask, "Is one of them a boy?" She replies, yes. What are the chances of the other child being a boy too? Suppose that a woman gives birth to a son 50% of the time and a daughter 50% of the time.
3) Suppose you know another woman who also has exactly two children. You do not know the gender of the younger one, but you know that the older child is a boy. What are the chances of the younger child being a boy too?
4a) Suppose in this country, there is a strange rule. women are to give birth, and they have to give birth until 1 and only 1 boy is born. This means that if a woman gives birth to a daughter, she will have to continue giving birth until she gives birth to a son, then she stops. Or if her first child is a boy, she stops giving birth. Suppose that a woman gives birth to a son 50% of the time and a daughter 50% of the time, what is the ratio of the number of boys to the number of girls after all the families have 1 son each? Do not factor in the number of men and women, only count the offsprings.
4b) The offspings grow up and get married, and they follow the same rule in the country. Then the new generation does the same. After many generations, what is the theoretical male to female ratio?
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If you have not tried those problems, please give yourself at least 5 minutes thinking through each problem. They are mathematical problems, there are no tricks involving words so pardon me for any bad english if any.
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(Answers coming out soon)
1) Suppose you are on a game show. There are 3 doors, one of them revealing a car behind it, or 1 million dollars, or anything you like. The other two have nothing behind them. You can only pick a door. So you pick a door, let's say door A. The host, who knows which door has the grand prize and which doors do not, then open one of the other 2 doors, either B or C, which has nothing. He will always open a door which has nothing behind it. So the question is, should you stick with your original choice, or should you switch, or it does not really matter? What are the chances of you winning the grand prize if you do not switch? Could it be 1 in 3, because your original choice was a 1 in 3 chance so whatever the host does, does not make a difference, or could it be 1 in 2, because now you have 2 choices, either your original door or the door that is left?
2) Suppose you know this woman who has exactly two children, but you do not know their genders. You ask, "Is one of them a boy?" She replies, yes. What are the chances of the other child being a boy too? Suppose that a woman gives birth to a son 50% of the time and a daughter 50% of the time.
3) Suppose you know another woman who also has exactly two children. You do not know the gender of the younger one, but you know that the older child is a boy. What are the chances of the younger child being a boy too?
4a) Suppose in this country, there is a strange rule. women are to give birth, and they have to give birth until 1 and only 1 boy is born. This means that if a woman gives birth to a daughter, she will have to continue giving birth until she gives birth to a son, then she stops. Or if her first child is a boy, she stops giving birth. Suppose that a woman gives birth to a son 50% of the time and a daughter 50% of the time, what is the ratio of the number of boys to the number of girls after all the families have 1 son each? Do not factor in the number of men and women, only count the offsprings.
4b) The offspings grow up and get married, and they follow the same rule in the country. Then the new generation does the same. After many generations, what is the theoretical male to female ratio?
---
If you have not tried those problems, please give yourself at least 5 minutes thinking through each problem. They are mathematical problems, there are no tricks involving words so pardon me for any bad english if any.
---
(Answers coming out soon)
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