Here are the answers to the previous post.
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1) The answer is that switching will ensure a 2 in 3 chance of winning the grand prize. The key thing here is that the host KNOWS which door contains the grand prize, so whatever door he opens will definitely be the door with nothing behind. So you have 2/3 chance of choosing a wrong door initially. Of the remaining 2 doors, he will open the remaining wrong door, so switching will definitely get you the grand prize. However, in 1/3 of the cases, you may choose the correct door initially. The host can open either of the remaining doors, but it does not matter, if you switch, you are going to select the wrong door. So switching will fail you only 1/3 of the time.
If you are still not convinced, consider this variation: You have to choose 1 door out of 1000 doors, in which only 1 of the doors has the grand prize behind it. You select one, the host opens 998 other doors which do not have the prize behind them. So you are left with your own door and another door which he did not open. Are you going to switch?
2) The chances of the other being a boy too is 1/3. Consider her having 2 kids, so 25% of the time she will have 2 girls, 25% 2 boys, and 50% 1 girl 1 boy (GG, GB, BG, BB). We discard GG because we know that at least one of them is a boy. As for the other 3 cases which have equal probability of happening (GB, BG, BB), only 1 of the cases means that the other child is a boy too.
3) This one is easy. You know that there are 2 kids, an older and a younger one. The gender of the younger one is not influenced by the gender of the older one. So the answer is 50%.
4) This is my favourite. The answer to part a is 1:1 ratio of boys to girls. Subsequently, the answer to part b is also 1:1. This is due to the fact that a woman gives birth to a boy 50% of the time (and a girl 50% of the time too)
You can consider this ratio because you only determine the number of children in that household. There can be one boy only, or there can be 100 girls with 1 boy. You are not changing the gender of any children, nor are you killing boys. If the question were to be changed to: girls are killed 50% of the time... then of course the answer will be different because you already determined that some girls will be killed (and not boys). But this is homicide so maybe not a good example, but the idea is there.
If you are still not convinced, consider an infinite number of children standing in a row. A household will pick children first in the row until it reaches a boy, in which it will accept. The next household will pick children after that doing the same thing. Of course some households will end up with 1 boy only, with no girls. The boy:girl ratio is hence 1:1.
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1) The answer is that switching will ensure a 2 in 3 chance of winning the grand prize. The key thing here is that the host KNOWS which door contains the grand prize, so whatever door he opens will definitely be the door with nothing behind. So you have 2/3 chance of choosing a wrong door initially. Of the remaining 2 doors, he will open the remaining wrong door, so switching will definitely get you the grand prize. However, in 1/3 of the cases, you may choose the correct door initially. The host can open either of the remaining doors, but it does not matter, if you switch, you are going to select the wrong door. So switching will fail you only 1/3 of the time.
If you are still not convinced, consider this variation: You have to choose 1 door out of 1000 doors, in which only 1 of the doors has the grand prize behind it. You select one, the host opens 998 other doors which do not have the prize behind them. So you are left with your own door and another door which he did not open. Are you going to switch?
2) The chances of the other being a boy too is 1/3. Consider her having 2 kids, so 25% of the time she will have 2 girls, 25% 2 boys, and 50% 1 girl 1 boy (GG, GB, BG, BB). We discard GG because we know that at least one of them is a boy. As for the other 3 cases which have equal probability of happening (GB, BG, BB), only 1 of the cases means that the other child is a boy too.
3) This one is easy. You know that there are 2 kids, an older and a younger one. The gender of the younger one is not influenced by the gender of the older one. So the answer is 50%.
4) This is my favourite. The answer to part a is 1:1 ratio of boys to girls. Subsequently, the answer to part b is also 1:1. This is due to the fact that a woman gives birth to a boy 50% of the time (and a girl 50% of the time too)
You can consider this ratio because you only determine the number of children in that household. There can be one boy only, or there can be 100 girls with 1 boy. You are not changing the gender of any children, nor are you killing boys. If the question were to be changed to: girls are killed 50% of the time... then of course the answer will be different because you already determined that some girls will be killed (and not boys). But this is homicide so maybe not a good example, but the idea is there.
If you are still not convinced, consider an infinite number of children standing in a row. A household will pick children first in the row until it reaches a boy, in which it will accept. The next household will pick children after that doing the same thing. Of course some households will end up with 1 boy only, with no girls. The boy:girl ratio is hence 1:1.
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