Probability Fun

pdurrant · 06-09-2010, 04:45 PM

Here's a fun bit of probability calculation. Taken (with some changes to make me happier about the answer) from an article in the New Scientist magazine.

Dan and Nick are talking to each other. Nick has no prior knowledge of Dan or his family. Work out the probability, at each numbered stage of the conversation, that Dan has exactly two male children (BB) given the information to that stage. Assume that the probability of any child of any parent being a boy is exactly 1 in 2.

Dan: I have exactly two children
1
Nick: Do you have at least one boy?
Dan: Yes.
2
Nick: Do you have at least one boy born on a Tuesday?
Dan: Yes.
3

I'll give the answers and reasoning behind the answers after there have been some comments. I will say that the probabilities at 1, 2 and 3 are all different.

I just hope that some other people on Mobileread are also a bit geeky about probability

Bilbo1967 · 06-09-2010, 04:53 PM

Interesting post - thanks.

I guess that possible answers are:

1. Possible combinations are; BB, BG, GG. So the probability of it being BB is 1/3

2. If there is at least one boy, then the possible options are BB and BG. The probability in this case is 1/2 for me.

3. I have to admit that I am not quite geeky enough for this one - I'm struggling to see the relevance of the birth day of the child. I'm guessing it makes it more likely that they're two boys though? I'm sure somebody will enlighten me shortly

omk3 · 06-09-2010, 05:28 PM

Hmm.
1) I'd say it's 1/4. It's 1/2 that the first child is a boy, and then another 1/2 that the second child is also a boy.
2)We discard the possibility that the first and second child are both girls, so we are left with three options, thus 1/3.
3) Erm... I've got to think a bit more about this

LazyScot · 06-09-2010, 05:29 PM

Hmmm.. I shall have to read my New Scientist. Which edition?

Quote:

Originally Posted by Bilbo1967

Interesting post - thanks.

I guess that possible answers are:

1. Possible combinations are; BB, BG, GG. So the probability of it being BB is 1/3

I think the combinations are BB BG GB GG, i.e. there are two "orders" that you can get one boy, so the chance of BB is 1/4....

Quote:

2. If there is at least one boy, then the possible options are BB and BG. The probability in this case is 1/2 for me.

I think I agree. Basically at this point the question is the same as the probability of one child being a boy.

Quote:

3. I have to admit that I am not quite geeky enough for this one - I'm struggling to see the relevance of the birth day of the child. I'm guessing it makes it more likely that they're two boys though? I'm sure somebody will enlighten me shortly

I can't help but feel that this is like the probablistic analysis of what you should do when you are given three choices, you choose one, and the one of the remaining two is taken away. Should you change you choice? The answer is yes.

I shall sleep on this one.

And I shall blame you for a sleepless night.....

(

)

omk3 · 06-09-2010, 05:33 PM

Oh great. Three people, three different answers, and that's only for the first two, 'simpler' stages.

obs20 · 06-09-2010, 05:40 PM

The chance of 2 boys
At stage 1 is 1/4 BB BG GB GG

At stage 2 the chance is now 1/2 either BG or BB

At stage 3 it gets tricky. The chance of having either child born on a Tuesday is 2/7, of one of those children being a boy is 1/7. The chance of any particular child being a boy born on a Tuesday is 1/14. That being said we still are in the same predicament as stage 2 with one of the children being a boy. So I guess the odds at this stage are still 1/2.

LazyScot · 06-09-2010, 05:58 PM

Quote:

Originally Posted by omk3

Oh great. Three people, three different answers, and that's only for the first two, 'simpler' stages.

Actually, I think this moving us on.

The first answer of 1/4, I'm confident of, since that is based on the importance of order (something tells me this is related to the difference between permutations and combinations....)

Anyhow, my first attempt at the second answer ignored order, and I think that was wrong, whereas yours did take it into account. Thus I think that the second answer should be 1/3 as the possibilities are BB BG GB.

If we were to expand the possibilities up each pair would come out to 49 possibilities. B(monday)B(monday), B(monday)B(tuesday), etc. However, we know that at least one boy was born on a tuesday, so the BG can only be expanded up to 7 possibilities each. B(tuesday)G(monday) and so on. By comparison, the BB expands up to 14 possibilities. So we have a total of 28 possibilities.

So the answer to 3 is, I think 1/2. (i.e. 14 out of 28)

omk3 · 06-09-2010, 06:10 PM

Hmmm

We know at least one boy was born on Tuesday.
So we have the options:

Girl (any day) - Boy (Tuesday)
Boy (Tuesday) - Girl (any day)
Boy (Tuesday) - Boy (any day)
Boy (any day) - Boy (Tuesday)
Boy (Tuesday) - Boy (Tuesday)

I think the last probability is 3/5, or 0,6.

beppe · 06-10-2010, 01:26 AM

Without any information we have

2B = 0.25
BG+GB= 0.5
GG=0.25
(the second is independent from the first one so we can add)

with the knowlege that one at least is a boy
we have to take away GG and renormalize to 1 so I divide for =.75. This gets
BB=1/3
BG+GB =2/3
GG=0

Tuesday has nothing to do with it (it would have been like asking is one of your boys left handed? or with blue eyes, or sleeps at night?)
so the probabilities remain the same.

Sparrow · 06-10-2010, 01:42 AM

Could it be something to do with the odds of Nick guessing correctly about a boy being born on a Tuesday?

pdurrant · 06-10-2010, 02:28 AM

Quote:

Originally Posted by beppe

Without any information we have

2B = 0.25
BG+GB= 0.5
GG=0.25
(the second is independent from the first one so we can add)

with the knowlege that one at least is a boy
we have to take away GG and renormalize to 1 so I divide for =.75. This gets
BB=1/3
BG+GB =2/3
GG=0

Tuesday has nothing to do with it (it would have been like asking is one of your boys left handed? or with blue eyes, or sleeps at night?)
so the probabilities remain the same.

Your analysis of the first two probabilities is spot on. But you've missed something for the third.

pdurrant · 06-10-2010, 02:28 AM

Quote:

Originally Posted by Sparrow

Could it be something to do with the odds of Nick guessing correctly about a boy being born on a Tuesday?

And you have correctly spotted the important point for working out the probability for the third answer.

pdurrant · 06-10-2010, 02:29 AM

Quote:

Originally Posted by LazyScot

Actually, I think this moving us on.

The first answer of 1/4, I'm confident of, since that is based on the importance of order (something tells me this is related to the difference between permutations and combinations....)

Anyhow, my first attempt at the second answer ignored order, and I think that was wrong, whereas yours did take it into account. Thus I think that the second answer should be 1/3 as the possibilities are BB BG GB.

If we were to expand the possibilities up each pair would come out to 49 possibilities. B(monday)B(monday), B(monday)B(tuesday), etc. However, we know that at least one boy was born on a tuesday, so the BG can only be expanded up to 7 possibilities each. B(tuesday)G(monday) and so on. By comparison, the BB expands up to 14 possibilities. So we have a total of 28 possibilities.

So the answer to 3 is, I think 1/2. (i.e. 14 out of 28)

You are very close, but not quite right on the third answer.

pdurrant · 06-10-2010, 02:32 AM

Quote:

Originally Posted by omk3

Hmmm

We know at least one boy was born on Tuesday.
So we have the options:

Girl (any day) - Boy (Tuesday)
Boy (Tuesday) - Girl (any day)
Boy (Tuesday) - Boy (any day)
Boy (any day) - Boy (Tuesday)
Boy (Tuesday) - Boy (Tuesday)

I think the last probability is 3/5, or 0,6.

On the right lines, but not right yet.

pdurrant · 06-10-2010, 02:34 AM

Quote:

Originally Posted by LazyScot

Hmmm.. I shall have to read my New Scientist. Which edition?

Umm... Well, not this week's (since I haven't got it yet) and not last week's. I think the one before that.

If you have a subscription you can read it on-line here:

http://www.newscientist.com/article/...html?full=true

NB. Given the way the question is posed in the article, I disagree with their answer.

06-09-2010, 04:45 PM	#1
pdurrant The Grand Mouse 高貴的老鼠 Posts: 73,860 Karma: 315126578 Join Date: Jul 2007 Location: Norfolk, England Device: Kindle Oasis	Probability Fun Here's a fun bit of probability calculation. Taken (with some changes to make me happier about the answer) from an article in the New Scientist magazine. Dan and Nick are talking to each other. Nick has no prior knowledge of Dan or his family. Work out the probability, at each numbered stage of the conversation, that Dan has exactly two male children (BB) given the information to that stage. Assume that the probability of any child of any parent being a boy is exactly 1 in 2. Dan: I have exactly two children 1 Nick: Do you have at least one boy? Dan: Yes. 2 Nick: Do you have at least one boy born on a Tuesday? Dan: Yes. 3 I'll give the answers and reasoning behind the answers after there have been some comments. I will say that the probabilities at 1, 2 and 3 are all different. I just hope that some other people on Mobileread are also a bit geeky about probability Last edited by pdurrant; 06-10-2010 at 10:35 AM. Reason: Fixed the setup to make my answer correct!

06-09-2010, 04:53 PM	#2
Bilbo1967 Not scared! Posts: 13,424 Karma: 81011643 Join Date: Mar 2009 Location: Midlands, UK Device: Kindle Paperwhite 10, Huawei M5 10	Interesting post - thanks. I guess that possible answers are: 1. Possible combinations are; BB, BG, GG. So the probability of it being BB is 1/3 2. If there is at least one boy, then the possible options are BB and BG. The probability in this case is 1/2 for me. 3. I have to admit that I am not quite geeky enough for this one - I'm struggling to see the relevance of the birth day of the child. I'm guessing it makes it more likely that they're two boys though? I'm sure somebody will enlighten me shortly Last edited by Bilbo1967; 06-09-2010 at 05:01 PM.

06-10-2010, 01:26 AM	#9
beppe Grand Sorcerer Posts: 5,161 Karma: 81026524 Join Date: Feb 2010 Location: Italy Device: Kindle3, Ipod4, IPad2	Without any information we have 2B = 0.25 BG+GB= 0.5 GG=0.25 (the second is independent from the first one so we can add) with the knowlege that one at least is a boy we have to take away GG and renormalize to 1 so I divide for =.75. This gets BB=1/3 BG+GB =2/3 GG=0 Tuesday has nothing to do with it (it would have been like asking is one of your boys left handed? or with blue eyes, or sleeps at night?) so the probabilities remain the same. Last edited by beppe; 06-10-2010 at 01:29 AM. Reason: adding reason for adding

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06-09-2010, 05:28 PM	#3
omk3 Wizard Posts: 1,454 Karma: 37243 Join Date: Dec 2009 Location: Europe Device: pocketbook 360, kindle 4	Hmm. 1) I'd say it's 1/4. It's 1/2 that the first child is a boy, and then another 1/2 that the second child is also a boy. 2)We discard the possibility that the first and second child are both girls, so we are left with three options, thus 1/3. 3) Erm... I've got to think a bit more about this

06-09-2010, 05:33 PM	#5
omk3 Wizard Posts: 1,454 Karma: 37243 Join Date: Dec 2009 Location: Europe Device: pocketbook 360, kindle 4	Oh great. Three people, three different answers, and that's only for the first two, 'simpler' stages.

06-09-2010, 05:40 PM	#6
obs20 Wizard Posts: 2,793 Karma: 29028512 Join Date: May 2010 Location: Florida Device: Sony PRS 600, Nook ST, Toshiba Excite 10.1 AT 300	The chance of 2 boys At stage 1 is 1/4 BB BG GB GG At stage 2 the chance is now 1/2 either BG or BB At stage 3 it gets tricky. The chance of having either child born on a Tuesday is 2/7, of one of those children being a boy is 1/7. The chance of any particular child being a boy born on a Tuesday is 1/14. That being said we still are in the same predicament as stage 2 with one of the children being a boy. So I guess the odds at this stage are still 1/2.

06-09-2010, 06:10 PM	#8
omk3 Wizard Posts: 1,454 Karma: 37243 Join Date: Dec 2009 Location: Europe Device: pocketbook 360, kindle 4	Hmmm We know at least one boy was born on Tuesday. So we have the options: Girl (any day) - Boy (Tuesday) Boy (Tuesday) - Girl (any day) Boy (Tuesday) - Boy (any day) Boy (any day) - Boy (Tuesday) Boy (Tuesday) - Boy (Tuesday) I think the last probability is 3/5, or 0,6.

06-10-2010, 01:42 AM	#10
Sparrow Wizard Posts: 4,395 Karma: 1358132 Join Date: Nov 2007 Location: UK Device: Palm TX, CyBook Gen3	Could it be something to do with the odds of Nick guessing correctly about a boy being born on a Tuesday?