Help with very counterintuitive problem

Dan Akers · science forum addict Joined: 19 Jul 2005 Posts: 56

Mike wrote;
"Of course it matters. The only way that 2/3 is the answer is if you
assume that the King can't / doesn't have an older brother (as in
primogeniture). Otherwise, the answer is most definitely 1/2 as your
intuition should tell you.
If you don't agree would love to hear your answer to Jon Haugsand's (and
the original poser's [Footer]) question. "
___________________________________
Re;
For any non-believers, try this simple experiment. You will need one
other person, a blind, and two coins. Have the other person flip the
two coins behind the blind. The person is instructed to show you one of
the coins that came up heads while keeping the other hidden. If both
coins turn up tails, then they are both reflipped; no guess is made.
When the person behind the blind shows you his selected coin, which will
be on heads, you then guess as to what the status of the other coin is.
If you guess tails every time, you will be right 2/3 of the time.
You could also accomplish the same thing by just flipping two coins over
and over again. Each time you get the H and H combination you jot down
an H. Each time you get the H and T combination you jot down a T. If
you get the T and T combination you jot down nothing and reflip. You
will find, if the coins are fair, that after 50 flips or so, you will
have jotted down about twice as many T's as H's.

-Dan Akers

Mike · science forum addict Joined: 17 Sep 2005 Posts: 74

Thanks, Dan, for a very thoughtful response but I don't agree. Your
experiment (as you describe it) will produce the result you say (2/3) but
the experiment does not capture the nature of the problem. The H and H
combination is as likely to produce a King as the H and T combination (if
you ignore primogeniture). If you assume that only the oldest male can be
King then I agree that the answer is 2/3 but you and other posters seem to
think the answer is 2/3 whether or not primogeniture applies.

Using the (1st born, 2nd born) notation from my earlier post, families with
two children can be represented as (b,b), (b,g), (g,b) and (g,g) and (we
assume) are equally likely. I guess we also agree that (g,g) families have
no chance of producing a King and we are thus left with the three other
combinations. Now, ask yourself what is the chance that each of these
three combinations will produce a King. Would you not say that a (b,b)
family has twice the chance of producing a King than either of the (g,b) or
(b,g) families alone or the same chance as families with the b and g
combination? Would not EACH boy in the (b,b) family have the same chance
as the (only) boy in each of the (b,g) or (g,b) families? If so, the answer
is 1/2.
If you redesign your coin-flipping experiment to operate on individuals
rather than on families, you'll get it.

"Dan Akers" <digikey@webtv.net> wrote in message
news:27829-433E9EB7-566@storefull-3136.bay.webtv.net...

Dan Akers · science forum addict Joined: 19 Jul 2005 Posts: 56

Mike wrote:
"Thanks, Dan, for a very thoughtful response but I don't agree. Your
experiment (as you describe it) will produce the result you say (2/3)
but the experiment does not capture the nature of the problem. The H and
H combination is as likely to produce a King as the H and T combination
(if you ignore primogeniture). If you assume that only the oldest male
can be King then I agree that the answer is 2/3 but you and other
posters seem to think the answer is 2/3 whether or not primogeniture
applies.
Using the (1st born, 2nd born) notation from my earlier post, families
with two children can be represented as (b,b), (b,g), (g,b) and (g,g)
and (we assume) are equally likely. I guess we also agree that (g,g)
families have no chance of producing a King and we are thus left with
the three other combinations. Now, ask yourself what is the chance
that each of these three combinations will produce a King. Would you not
say that a (b,b) family has twice the chance of producing a King than
either of the (g,b) or (b,g) families alone or the same chance as
families with the b and g combination? Would not EACH boy in the
(b,b) family have the same chance as the (only) boy in each of the (b,g)
or (g,b) families? If so, the answer is 1/2.
If you redesign your coin-flipping experiment to operate on individuals
rather than on families, you'll get it."
____________________________________
Re;
Hi Mike... Sorry, but I disagree. The fact that the individual IS a
king establishes that he is a male and for the sake of argument, we can
say that he is also the eldest son if he happens to have a brother. We
also know that he is one of two children. The question is: What is the
probability that the other sibling is a female. This interrogative
makes the fact that he may or may not be the eldest son mute in this
problem. In the case of querying about the probability of a sister, it
doesn't matter if she or he is the elder (as far as I know that has no
bearing on him being a king).
I think my coin analogy is dead on. The person behind the blind
flipping coins is analogous to the person introducing the king. The
coins represent the two royal children, in this case the king and his
anonymous sibling. The manipulation of always presenting a "head" from
the flipped result is analogous to the fact that the "introducer" has
introduced a male from a pair of siblings.
Even if you ask the question: "What is the probability that the sibling
is a male and younger?", the answer is still 1/3 (as in my example
problem; P(BB) in two children given one is a male, from an earlier
post). The reason being, that the order of birth does not affect the
probability of the permutation BB.
However, if you asked" What is the probability of the sibling being a
sister AND younger?", then the order does have bearing and the
probability reduces from 2/3 to 1/3. Keep in mind, we are talking about
the probability of particular permutations not combinations; BG does not
equal GB in the younger sister case, but they are equal in the older OR
younger sister case, which is the problem of the OP.

-Dan Akers

Dan Akers · science forum addict Joined: 19 Jul 2005 Posts: 56

Mike wrote:
"Your experiment (as you describe it) will produce the result you say
(2/3) but the experiment does not capture the nature of the problem. The
H and H combination is as likely to produce a King as the H and T
combination (if you ignore primogeniture)."
____________________________________
Re;
That is the essence of the problem; the counterintuitive land-mine; the
combination HH is not probabilistically equivalent to the TH combination
in this problem. The reason being, that the TH combo has two
permutations, namely TH and HT. The HH combination has only one
permutation.
______________________________________
MIke wrote;
"If you assume that only the oldest male can be King then I agree that
the answer is 2/3 but you and other posters seem to think the answer is
2/3 whether or not primogeniture applies."
______________________________________
Re;
Yes, that is how I see it. As an earlier poster said, it doesn't matter
about the birth order in this problem. The age of the king relative to
the sister or brother does not have bearing on the problem since it
doesn't matter if the king is older or younger than the sister, ie P(BG
or GB)=2/3 when the sister can be younger or older. Even in the case of
a brother, it does not matter that the king is required to be the elder
of the two; P(BB) does not change with this condition; it is still 1/3.

-Dan Akers

Mike · science forum addict Joined: 17 Sep 2005 Posts: 74

Dan wrote (in another post):
"The first condition on the group, to make it an analogy of the "king"
problem, would be to make an announcement at the gathering that all
girl-girl or GG pairs are to please leave the gathering. Since this
eliminates one of the four unique possible permutational possibilities,
only BG, GB, and BB remain. Thus, on a random selection of boys, 2/3
would say they were with a sister and 1/3 with a brother."

Re;
For you to make the last statement you are in effect assuming primogeniture
in which case I fully agree with you. On the other hand since you don't
explicitly say anything about order of succession I would assume you are
allowing either boy in a BB family to become King in which case I fully
disagee with you. Referring to your problem description above if, after the
GG pairs left the gathering, you mixed all of the remaining boys and girls
represented in the three remaining family configurations BG, GB and BB (each
family configuration equally represented) then a boy polled at random from
the entire population would be as likely say that he is with a brother as
with a sister. That's because there are as many boys from BB families as
there are boys from BG AND GB families or, equivalently, twice as many boys
from BB families as there are from either BG OR GB families.

"Dan Akers" <digikey@webtv.net> wrote in message
news:9587-433EFAD4-642@storefull-3134.bay.webtv.net...

Dan Akers · science forum addict Joined: 19 Jul 2005 Posts: 56

I wrote;
"The first condition on the group, to make it an analogy of the "king"
problem, would be to make an announcement at the gathering that all
girl-girl or GG pairs are to please leave the gathering. Since this
eliminates one of the four unique possible permutational possibilities,
only BG, GB, and BB remain. Thus, on a random selection of boys, 2/3
would say they were with a sister and 1/3 with a brother."
____________________________________
To wit Mike replied:
"For you to make the last statement you are in effect assuming
primogeniture in which case I fully agree with you. On the other hand
since you don't explicitly say anything about order of succession I
would assume you are allowing either boy in a BB family to become King
in which case I fully disagee with you. Referring to your problem
description above if, after the GG pairs left the gathering, you mixed
all of the remaining boys and girls represented in the three remaining
family configurations BG, GB and BB (each family configuration equally
represented) then a boy polled at random from the entire population
would be as likely say that he is with a brother as with a sister.
That's because there are as many boys from BB families as there are boys
from BG AND GB families or, equivalently, twice as many boys from BB
families as there are from either BG OR GB families."
___________________________________
Re;
You're right. The statement that I made about the gathering of sibling
pairs, namely; "on a random selection of boys, 2/3 would say they were
with a sister and 1/3 with a brother" is incorrect and in fact should
say, "1/2 would say they were with a sister and 1/2 with a brother".
This would seem to prove your argument. I will have to give it further
thought and see if I can reconcile this apparent fact with the coin
analogy. I'm not entirely sure right now that the "gathering" is wholly
analogous to the OP's original problem. By that I mean, kings and
queens aside, that the gathering analogy seems to contradict every
"given that one of two siblings is a boy, what is the probability that
the other is a girl?" problem that I have encountered.

-Dan Akers

Mike · science forum addict Joined: 17 Sep 2005 Posts: 74

Thanks, Dan. It would indeed be interesting to see the problem exactly as
it was posed in Footer's textbook to resolve lingering doubts about the
answer. In most probability textbooks "real world" assumptions are usually
explicitly stated (not to mention greatly simplified) which makes me doubt
that the book had intended the student to assume primogeniture. On the
other hand, the book did say "King" (as opposed to merely saying named
"John" or "Joe") and gave the answer as 2/3 in which case I guess the author
intended students to make a (reasonable?) inference that primogeniture
applied.

"Dan Akers" <digikey@webtv.net> wrote in message
news:27830-433F285D-170@storefull-3136.bay.webtv.net...

Nigel · science forum beginner Joined: 03 Jun 2005 Posts: 37

Mike wrote:

Footer · science forum beginner Joined: 24 Sep 2005 Posts: 2

Mike wrote:
"It would indeed be interesting to see the problem exactly as it was
posed in Footer's textbook to resolve lingering doubts about the
answer."

The original question was taken verbatim from the text.

Thanks for the responses (I never expected this much of a debate)!
After reading through the thread and thinking about this some more, it
seems to me that there are two separate questions here (with different
answers) that are being interchanged:

1) What is the probability that a random sibling pair contains a girl,
given that it contains at least one boy?
2) What is the probability that a random boy comes from a boy-girl
pair, given that he has one sibling?

The difference between these two is that #1 concerns a pair and #2
concerns a single boy. The answer to #1 is most definitely 2/3, and I
think is analogous to Dan's coin experiment (which I have no doubt is
correct). However, the answer to #2 is 1/2. We can keep the same sample
space (BB, BG, GB), but I think the trick is to realize that now we are
selecting a _boy_ at random, whereas in #1 we were selecting a _pair_
at random. So of the four boys listed above, two have a sister and two
have a brother. I would argue that example #2 is analogous to the
"gathering" scenario and to the original problem, making the answer in
the book incorrect.

Unless we were supposed to take primogeniture into account, in which
case I don't care anymore.

Dan Akers · science forum addict Joined: 19 Jul 2005 Posts: 56

Footer wrote;
"After reading through the thread and thinking about this some more, it
seems to me that there are two separate questions here (with different
answers) that are being interchanged:
1) What is the probability that a random sibling pair contains a girl,
given that it contains at least one boy?
2) What is the probability that a random boy comes from a boy-girl pair,
given that he has one sibling?
The difference between these two is that #1 concerns a pair and #2
concerns a single boy. The answer to #1 is most definitely 2/3, and I
think is analogous to Dan's coin experiment (which I have no doubt is
correct). However, the answer to #2 is 1/2. We can keep the same sample
space (BB, BG, GB), but I think the trick is to realize that now we are
selecting a _boy_ at random, whereas in #1 we were selecting a _pair_ at
random. So of the four boys listed above, two have a sister and two have
a brother. I would argue that example #2 is analogous to the "gathering"
scenario and to the original problem, making the answer in the book
incorrect."
_____________________________________
Re;
How about considering the gathering in this way... Imagine that the
room contains a large population of sibling pairs; say 4000 pairs. Of
these it is expected that 1000 will be BB, 1000 GG, 1000 BG, and 1000
GB. Now we ask the GG pairs to leave. Then we place a curtain down the
center of the room. We then ask that the pairs spilt on either side of
the curtain, with the stipulation that only boys can be on the right
side of the curtain. Then, from the 3000 pairs there will be 3000 boys
on the right side of the curtain while on the left side there will be
1000 boys and 2000 girls. Now we poll the boys on the right as to
whether or not they have a sister and we, of course, find that 2/3 have
sisters.
This "selection" of a male of the pair, in my mind at least, makes the
difference from polling any random boy. I think this is the case in the
original problem with the King; namely a male from a known pair is
"selected" to be introduced; what is the probability of his having a
sister? I would have to say the book is correct, namely 2/3. The fact
that he is a "king" is irrelevant, except to establish the fact that he
is male because even if birth order were important in the case of BB
pairs, it would not change the outcome of the polling.

-Dan Akers

Jon Haugsand · science forum beginner Joined: 03 May 2005 Posts: 37

* Dan Akers

Mike · science forum addict Joined: 17 Sep 2005 Posts: 74

Dan wrote:

Clinton · science forum beginner Joined: 10 Sep 2005 Posts: 46

Mike wrote:

Mike · science forum addict Joined: 17 Sep 2005 Posts: 74

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