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Probability Problems


ChumpusRex

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Inspired by the discussions in the premium bond vs lottery thread, and the germane point made in that thread:

However, probabilites are weird.

Here are a few questions for people to chew the cud over:

1. Alice has a child. What is the probability that the child is a boy?

2. Alice has 2 children. She tells you that one of them is a boy. What is the probability that the other child is a boy?

3. Alice has 2 children. She tells you that one of them is a boy, born on a Monday. What is the probability that the other child is a boy?

Explanatory notes: You should assume that there is a perfectly even geneder split, and ignore the possibility of ambiguous gender.

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Inspired by the discussions in the premium bond vs lottery thread, and the germane point made in that thread:

Here are a few questions for people to chew the cud over:

1. Alice has a child. What is the probability that the child is a boy?

2. Alice has 2 children. She tells you that one of them is a boy. What is the probability that the other child is a boy?

3. Alice has 2 children. She tells you that one of them is a boy, born on a Monday. What is the probability that the other child is a boy?

Explanatory notes: You should assume that there is a perfectly even geneder split, and ignore the possibility of ambiguous gender.

1 Depends whether Alice is from one of those cultures that so desperately overvalues boys they have early gender tests and selectively abort the girls.

2 Ditto

3 Ditto

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Inspired by the discussions in the premium bond vs lottery thread, and the germane point made in that thread:

Here are a few questions for people to chew the cud over:

1. Alice has a child. What is the probability that the child is a boy?

2. Alice has 2 children. She tells you that one of them is a boy. What is the probability that the other child is a boy?

3. Alice has 2 children. She tells you that one of them is a boy, born on a Monday. What is the probability that the other child is a boy?

Explanatory notes: You should assume that there is a perfectly even geneder split, and ignore the possibility of ambiguous gender.

Kind of depends how you interpret 'one of them is a boy'. This might be taken to mean specifically that one was a boy, no more, no less, i.e not two, not Zero, in which case the probability of the second one being a boy would be zero.

1) .5

2) 0

3) 0

Maybe I've had too much beer.

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Inspired by the discussions in the premium bond vs lottery thread, and the germane point made in that thread:

Here are a few questions for people to chew the cud over:

1. Alice has a child. What is the probability that the child is a boy?

2. Alice has 2 children. She tells you that one of them is a boy. What is the probability that the other child is a boy?

3. Alice has 2 children. She tells you that one of them is a boy, born on a Monday. What is the probability that the other child is a boy?

Explanatory notes: You should assume that there is a perfectly even geneder split, and ignore the possibility of ambiguous gender.

1. 50/50 (roughly)

2. possible combos. 2 boys, 2 x boy and girl or 2 x girl. so 1/4

3. I would have though same as 2.

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Inspired by the discussions in the premium bond vs lottery thread, and the germane point made in that thread:

Here are a few questions for people to chew the cud over:

1. Alice has a child. What is the probability that the child is a boy?

2. Alice has 2 children. She tells you that one of them is a boy. What is the probability that the other child is a boy?

3. Alice has 2 children. She tells you that one of them is a boy, born on a Monday. What is the probability that the other child is a boy?

Explanatory notes: You should assume that there is a perfectly even geneder split, and ignore the possibility of ambiguous gender.

.5 .5 .5

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Here are a few questions for people to chew the cud over:

1. Alice has a child. What is the probability that the child is a boy?

2. Alice has 2 children. She tells you that one of them is a boy. What is the probability that the other child is a boy?

3. Alice has 2 children. She tells you that one of them is a boy, born on a Monday. What is the probability that the other child is a boy?

Explanatory notes: You should assume that there is a perfectly even geneder split, and ignore the possibility of ambiguous gender.

1. 0.5

2. 1/3

3. 1/3

But I love finding out I'm wrong with probabilities, as you say they are weird, or more technically counter-intuitive.

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You seem to be also favouring the posterior probability route. My 0.375 was plucked out of the air as being the likely answer if you applied the posterior probability formula (although I couldn't actually be bothered to look it up again or do the maths).

I don't know what you are talking about and it cannot possibly be 0.375 unless you are answering a different question to that which was posed.

I believe all I used was conditional probability and excluded irrelevant information for part 3.

But I daren't go on for fear X-QUORK thinks me boring. Heaven forbid!

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The question is strangely familiar. Hmmm ... You should've subtitled this thread "who listens to more or less on the wireless?"

As it happens, I know this Alice, and her identical twins. Other posters in this thread have replied based on different readings of the question, and what exact information we've been given.

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Inspired by the discussions in the premium bond vs lottery thread, and the germane point made in that thread:

Here are a few questions for people to chew the cud over:

1. Alice has a child. What is the probability that the child is a boy?

0.5

2. Alice has 2 children. She tells you that one of them is a boy. What is the probability that the other child is a boy?

0.5 Information about one of them being a boy has no bearing on whether or not the other one is a boy as well. Each child is an independent event.

3. Alice has 2 children. She tells you that one of them is a boy, born on a Monday. What is the probability that the other child is a boy?

0.5 for the same reason as 2) the question was just more convoluted

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0.5 Information about one of them being a boy has no bearing on whether or not the other one is a boy as well. Each child is an independent event.

I'm afraid this is incorrect (but totally understandable). The new information allows you to rule out the case where there are two girls. This leaves 3 possible cases, only one of which is the two boy case, hence 1/3.

The question does not ask what are the odds of the next child being a boy but rather given that there are already 2 children and one is a boy what is the probability that the other is also a boy.

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Inspired by the discussions in the premium bond vs lottery thread, and the germane point made in that thread:

Here are a few questions for people to chew the cud over:

1. Alice has a child. What is the probability that the child is a boy?

2. Alice has 2 children. She tells you that one of them is a boy. What is the probability that the other child is a boy?

3. Alice has 2 children. She tells you that one of them is a boy, born on a Monday. What is the probability that the other child is a boy?

Explanatory notes: You should assume that there is a perfectly even geneder split, and ignore the possibility of ambiguous gender.

A significant part of the problem here comes from the choice of scenario (I don't think sex is completely random) and the wording of the questions rather than probability.

But, assuming that sex is completely random and independent, and that boys and girls are equally probable:

1. 50% that Alice's child will be a boy.

2. If you mean that Alice is saying that at least one of her children is a boy, then there's a 50% chance that Alice's other child is also a boy. (Probability is a temporal thing: it depends when you ask the question. In this case we're looking back at two children who have already been born, their sexes having been established, and the sex of one child is independent of the other. However, if you were to say, in advance of Alice having her two children, what's the probability that both of them will be boys, then the answer would be 25%. The temporal nature of probability -- depending at what point during proceedings you establish the probability of combination of events -- seems likely to be the reason for much of the confusion on this and the other thread.)

3. The way this one is worded, the Monday thing doesn't appear relevant. Perhaps you could re-word it to explain what the born-on-Monday issue is about. Are you saying that only one of Alice's two children was born or a Monday, or what are you saying?

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I'm afraid this is incorrect (but totally understandable). The new information allows you to rule out the case where there are two girls. This leaves 3 possible cases, only one of which is the two boy case, hence 1/3.

The question does not ask what are the odds of the next child being a boy but rather given that there are already 2 children and one is a boy what is the probability that the other is also a boy.

This has to be correct. I know I put 1/4 but this was my logic and I was pissed. :lol:

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I'm afraid this is incorrect (but totally understandable). The new information allows you to rule out the case where there are two girls. This leaves 3 possible cases, only one of which is the two boy case, hence 1/3.

The question does not ask what are the odds of the next child being a boy but rather given that there are already 2 children and one is a boy what is the probability that the other is also a boy.

I haven't slept yet, so forgive me if I'm being ultra stupid. How are there 3 possible cases?

We know the gender of 'Child1'. It Is a boy, so there are only two possible cases:

Child1: Child2

Boy: Boy

Boy: Girl

Girl: Boy - can't be as Child1 is a Boy

Girl:Girl - can't be as Child1 is a Boy

No?

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Inspired by the discussions in the premium bond vs lottery thread, and the germane point made in that thread:

Here are a few questions for people to chew the cud over:

1. Alice has a child. What is the probability that the child is a boy?

2. Alice has 2 children. She tells you that one of them is a boy. What is the probability that the other child is a boy?

3. Alice has 2 children. She tells you that one of them is a boy, born on a Monday. What is the probability that the other child is a boy?

Explanatory notes: You should assume that there is a perfectly even geneder split, and ignore the possibility of ambiguous gender.

1. 0.5

2. Alice is a bird. This means the probability of her making this whole thing up is very large. Therefore in all probability Alice has no children and instead has a cat called dipsy and a dildo called Dave.

3. Ditto.

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Inspired by the discussions in the premium bond vs lottery thread, and the germane point made in that thread:

Here are a few questions for people to chew the cud over:

1. Alice has a child. What is the probability that the child is a boy?

2. Alice has 2 children. She tells you that one of them is a boy. What is the probability that the other child is a boy?

3. Alice has 2 children. She tells you that one of them is a boy, born on a Monday. What is the probability that the other child is a boy?

Explanatory notes: You should assume that there is a perfectly even geneder split, and ignore the possibility of ambiguous gender.

Gut instinct says 0.5, 0.5 & 0.5 as I can't see any relevance to the extra information. However, I think the pertinent question is how many bedrooms does Alice's council flat have? :rolleyes:

edit: JustYield's answer makes sense to me. 0.5, 1/3 and 1/3 sounds good.

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I haven't slept yet, so forgive me if I'm being ultra stupid. How are there 3 possible cases?

We know the gender of 'Child1'. It Is a boy, so there are only two possible cases:

Child1: Child2

Boy: Boy

Boy: Girl

Girl: Boy - can't be as Child1 is a Boy

Girl:Girl - can't be as Child1 is a Boy

No?

Girl:boy is a possibility. One of the children is a boy, not necessarily child1.

edit: added "necessarily"

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Girl:boy is a possibility. One of the children is a boy, not child1.

Ah, yes. In my mind I was thinking older child and sibling.

So, I think the answer is:

1/2 (two possible outcomes)

1/3 (three possible outcomes)

1/4 (four possible outcomes)

Edit: Hmmm...... thinking about part 3, I'm not sure how to calculate the possible outcomes - really not sure about the last one. Excel is just shrugging it's shoulders at me ..... :unsure:

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I'm afraid this is incorrect (but totally understandable). The new information allows you to rule out the case where there are two girls. This leaves 3 possible cases, only one of which is the two boy case, hence 1/3.

The question does not ask what are the odds of the next child being a boy but rather given that there are already 2 children and one is a boy what is the probability that the other is also a boy.

Have fun on the roulette table mate... laugh.gif

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  • 429 Brexit, House prices and Summer 2020

    1. 1. Including the effects Brexit, where do you think average UK house prices will be relative to now in June 2020?


      • down 5% +
      • down 2.5%
      • Even
      • up 2.5%
      • up 5%



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