Probability Fun - Page 2

beppe · 06-10-2010, 02:45 AM

Quote:

Originally Posted by pdurrant

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

Bayesian adept!
That's OK.

pdurrant · 06-10-2010, 03:43 AM

Quote:

Originally Posted by pdurrant

Assume that the probability of any child of any parent being a boy is exactly 1 in 2.

I should also have said: assume that the chance of a child being born on a Tuesday is exactly 1 in 7

But it looks like everyone's assumed that anyway.

dreams · 06-10-2010, 03:45 AM

What a fun post, pdurrant.

The first two were pretty standard. It's the last one that made me think this was one of those trick probability questions.

The third answer is driving me crazy. It's a 1/7 probability that he guess correctly that the boy was born on a Tuesday. So that changes all the formulas for continuing to work out the answer.

I am looking forward to the reasoning for the answer.

pdurrant · 06-10-2010, 04:05 AM

Quote:

Originally Posted by dreams

The third answer is driving me crazy. It's a 1/7 probability that he guess correctly that the boy was born on a Tuesday.

The probability that Nick has guessed correctly is not 1 in 7, although the answer to the question you have given is indeed 1 in 7. The exact wording of Nick's question is important.

beppe · 06-10-2010, 04:12 AM

Quote:

Originally Posted by pdurrant

I should also have said: assume that the chance of a child being born on a Tuesday is exactly 1 in 7

But it looks like everyone's assumed that anyway.

You are so kind.
I still have doubts about the relevance of this.

Why there is nothing about the weight of the boy at the moment of birth. There is a 95% probability that it will be over 2.8 Kg. and 50% that it is over 3,6.

This seems to be more important for the well being of the child than being born under the influence of Mars (the god). I was born on Sunday (the laziest) Why this is not taken into account?

And the age of the captain?

Of course I am teasing. This is a nice thread. Better, it is a very nice thread.

Sparrow · 06-10-2010, 04:27 AM

All I can think of is the chances of guessing Tuesday right are 1 in 7 with 1 boy, and 2 in 7 with 2 boys.

dreams · 06-10-2010, 04:34 AM

Does this make the 1 / 7?

B (Tues) B (Tues) twins
B (Tues) B (Mon)
B (Tues) B (Wed)
B (Tues) B (Thurs)
B (Tues) B (Fri)
B (Tues) B (Sat)
B (Tues) B (Sun)

I probably shouldn't try to think so late at night.

pdurrant · 06-10-2010, 04:51 AM

Quote:

Originally Posted by dreams

Does this make the 1 / 7?

B (Tues) B (Tues) twins
B (Tues) B (Mon)
B (Tues) B (Wed)
B (Tues) B (Thurs)
B (Tues) B (Fri)
B (Tues) B (Sat)
B (Tues) B (Sun)

I probably shouldn't try to think so late at night.

You're making the same kind of mistake as those that think that the probability of having two boys when having two children is 1 in 3 because there are three combinations - BB BG and GG.

Let's used TB for a boy born on Tuesday ("Tuesday Boy") and OB for a boy born on another day ("Other Boy").

You've considered
TB TB and TB OB but you've neglected OB TB.

Oh - and TB TB don't have to be twins - they could both be born on a Tuesday, just different Tuesdays (in different years, usually!).

dreams · 06-10-2010, 05:03 AM

Quote:

You've considered
TB TB and TB OB but you've neglected OB TB.

You see, that is what gets to me. In reality, you either have a B as the next child or you don't, so it's a 1/2.

I really shouldn't try to reason when the alarm is set to go off in 4 hours. (I really want to know the answer, but will have to wait until I get home from work.)

pdurrant · 06-10-2010, 05:37 AM

Quote:

Originally Posted by dreams

You see, that is what gets to me. In reality, you either have a B as the next child or you don't, so it's a 1/2.

And if the question was "A couple have two children, what is the probability that their second child is a boy?" 1 in 2 is the right answer.

But that's /not/ the question. Nowhere is it specified which of the two children we're talking about.

omk3 · 06-10-2010, 05:59 AM

Ok next try.
We have three possible combinations, GB, BG, BB.
The chances of the boy being born on a Tuesday on each of the first two cases is 1/7.
In the BB case, we have 49 possible combinations, in 13 of which there is at least on Tuesday. So the chances of at least on boy being born on a Tuesday in a BB case are 13/49.
Which makes the overall probability....

...13/27?

Oh how I wish I remembered how to calculate these things...

pdurrant · 06-10-2010, 06:09 AM

Quote:

Originally Posted by omk3

Ok next try.
We have three possible combinations, GB, BG, BB.
The chances of the boy being born on a Tuesday on each of the first two cases is 1/7.

Yes.

Quote:

Originally Posted by omk3

In the BB case, we have 49 possible combinations, in 13 of which there is at least [one boy born] on Tuesday.
So the chances of at least one boy being born on a Tuesday in a BB case are 13/49.

Yes.

Quote:

Originally Posted by omk3

Which makes the overall probability....

You've got the right numbers so far. You just need to combine them with the probabilities of the two cases you've analysed, and then you'll have the answer...

pdurrant · 06-10-2010, 06:26 AM

Quote:

Originally Posted by omk3

Which makes the overall probability....

...13/27?

Oh how I wish I remembered how to calculate these things...

Yes! We have a winner! 13/27 is indeed the right answer.

I like probability trees myself. In the attached chart, the first branch (top down) is the sex of the first child, the next is whether the child (if a boy) was born on a tuesday. The third branchings are for the sex of the second child, and the last for whether that child (of a boy) was born on a Tuesday.

The probabilities across the bottom are the chance of getting to that outcome, normalised to fractions of 196 (= 2x7x2x7), and are obtained just by multiplying the probabilities of each branch that leads to that outcome.

Now, we know that Dan has at least one boy, so we can ignore the GG outcome. And we also know that at least one of his children is a boy born on a Tuesday. So we can now ignore the outcomes that don't include a Boy born on a Tuesday - that's the OBOB, OBG and GOB outcomes.

We're left with
TBTB - 1/196
TBOB - 6/196
TBG - 7/196
OBTB - 6/196
GTB - 7/196

The total probability of getting one of these outcomes of 27/196. But in only TBTB, TBOB and OBTB do we have two boys - 13/196.

So the probability of Dan having two boys, given that he has at least one boy and that Nick guessed a day correctly is 13 in 27.

omk3 · 06-10-2010, 06:32 AM

Quote:

Originally Posted by pdurrant

Yes! We have a winner! 13/27 is indeed the right answer.

We do?

Thank you for a very engaging puzzle!

I'm usually ok with logic puzzles, but I'm always nervous around probabilities.
I firmly believe that "a million-to-one chance succeeds nine times out of ten", so nothing makes much sense anyway.

Javed · 06-10-2010, 06:33 AM

Well done omk3; I'm definitely giving you karma for figuring that out.

06-10-2010, 05:59 AM	#26
omk3 Wizard Posts: 1,454 Karma: 37243 Join Date: Dec 2009 Location: Europe Device: pocketbook 360, kindle 4	Ok next try. We have three possible combinations, GB, BG, BB. The chances of the boy being born on a Tuesday on each of the first two cases is 1/7. In the BB case, we have 49 possible combinations, in 13 of which there is at least on Tuesday. So the chances of at least on boy being born on a Tuesday in a BB case are 13/49. Which makes the overall probability.... ...13/27? Oh how I wish I remembered how to calculate these things... Last edited by omk3; 06-10-2010 at 06:08 AM.

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06-10-2010, 03:45 AM	#18
dreams It's about the umbrella Posts: 25,110 Karma: 56250158 Join Date: Jan 2009 Device: Sony 505\| K Fire \| KK 3G+Wi-Fi \| iPhone 3Gs \|Vista 32-bit Hm Prem w/FF	What a fun post, pdurrant. The first two were pretty standard. It's the last one that made me think this was one of those trick probability questions. The third answer is driving me crazy. It's a 1/7 probability that he guess correctly that the boy was born on a Tuesday. So that changes all the formulas for continuing to work out the answer. I am looking forward to the reasoning for the answer.

06-10-2010, 04:27 AM	#21
Sparrow Wizard Posts: 4,395 Karma: 1358132 Join Date: Nov 2007 Location: UK Device: Palm TX, CyBook Gen3	All I can think of is the chances of guessing Tuesday right are 1 in 7 with 1 boy, and 2 in 7 with 2 boys.

06-10-2010, 04:34 AM	#22
dreams It's about the umbrella Posts: 25,110 Karma: 56250158 Join Date: Jan 2009 Device: Sony 505\| K Fire \| KK 3G+Wi-Fi \| iPhone 3Gs \|Vista 32-bit Hm Prem w/FF	Does this make the 1 / 7? B (Tues) B (Tues) twins B (Tues) B (Mon) B (Tues) B (Wed) B (Tues) B (Thurs) B (Tues) B (Fri) B (Tues) B (Sat) B (Tues) B (Sun) I probably shouldn't try to think so late at night.

06-10-2010, 06:33 AM	#30
Javed Grand Sorcerer Posts: 10,270 Karma: 1126878541 Join Date: Oct 2009 Device: Astak Pocket PRO, iPod Touch, PRS-650	Well done omk3; I'm definitely giving you karma for figuring that out.