Invalid Jeopardy question?
Rick Schubert
[Sorry if this is not the proper newsgroup for this question, but I thought
it might be since there has been a lot of discussion here lately about
Jeopardy. Also, I thought that the type of people who do post here might
be able to answer this question. Finally, the only other newsgroup I could
think of that might be more appropriate would be sci.bio, and this seemed
too elementary for them.]
Now with that out of the way:
The other night (I think it was 4/2), there was a Daily Double question
that I thought had no correct answer (or in Jeopardy terms, an answer that
had no correct question). It was a genetics question, that, to paraphrase,
asked what kind of hair the baby of a black, straight-haired father and a
blonde, curly-haired mother would "most likely" have. The answer given was
black, curly-haired.
I contend that there is no correct answer. I suspect that their reasoning
was that black and curly-haired are dominant. But, I contend that no matter
which are dominant, one cannot answer based on the information given.
We can know the genetic make-up of a parent that has a recessive trait
(both genes for that trait are recessive), but we cannot know the genetic
make-up of the parent with the dominant gene (we know one of the two is
dominant, but we do not know the other). Therefore, the parent with the
dominant trait does not necessarily pass on the dominant trait to the
baby, so we cannot know what trait the baby will have.
If we knew what percentage of black-haired men had 2 black haired genes,
etc., we could compute probabilities and find out what percentage of such
babies would have black hair, etc., so we could determine which combination
was the most likely of the 4 possibilities, but I don't think that that was
what the question was getting at.
Is my reasoning correct?
Ob. Jeopardy puzzle #1:
If you get the Daily Double in Double Jeopardy with time running out and
have $3000, with first place having $11900 and third place having $2500,
how much should you bet?
Ob. Jeopardy puzzle #2:
If you get the Daily Double in Double Jeopardy with time running out and
have $6000, with first place having $11900 and third place having $2500,
how much should you bet?
Ob. Jeopardy puzzle #3:
Why does Alex seem so surprised when someone wins by "only" $1?
-- Rick Schubert (r...@se-sd.sandiego.NCR.COM)
Raymond Chen
>What kind of hair the baby of a black, straight-haired father and a
>blonde, curly-haired mother would "most likely" have.
SPOILERS
I sent this to Rick, but I think it merits general distribution
(so that peopel can email me any dissenting opinions).
We follow the analysis of Dave Seaman (a...@seaman.cc.purdue.edu).
> 1. Black hair is dominant; blonde hair is recessive (quite
> reasonable).
>
> 2. Curly hair is dominant; straight hair is recessive (also quite
> reasonable).
I maintain that this (plus one more quite reasonable assumption) is all the
information we need.
We are all agreed that the mother must be homogeneous recessive for blonde
hair. The father, on the other hand, can be either homogeneous or
heterogenous for black hair. Let p be the probability that the father
is homogeneous.
ASSUMPTION: p > 0. That is, there exist SOMEBODY in the world that is
homogenous for black hair. A very reasonable assumption.
Now we all know that if the father is homogenous dominant for black hair,
then the probability of the child having black hair is 1, whereas if he
is heterogenous then the probability is 1/2.
So the total probability is
1 x p + 1/2 x (1-p) = 1/2 + 1/2 p
so assuming that p > 0 we see that the probability is > 1/2.
Now I made the simplifying assumption that the only other hair color
in question is blonde. If the father is heterogenous black/red, say,
that doesn't change the probability for the child's hair being black, but
it does lower the probability of the child's hair being blonde, so if
you didn't like the assumption that p>0, we could replace it with the
ALTERNATE ASSUMPTION: The probability that the father is heterogenous
for black and some hair color other than blonde is > 0.
DISCLAIMER: We all know that the mechanism for determination of hair
color is much more complicated than this. This is a puzzle,
not a genetics exam!
--
ray...@math.berkeley.edu mathematician by training, hacker by choice
What`s in a name?
>had no correct question). It was a genetics question, that, to paraphrase,
>asked what kind of hair the baby of a black, straight-haired father and a
>blonde, curly-haired mother would "most likely" have. The answer given was
>black, curly-haired.
>
>I contend that there is no correct answer. I suspect that their reasoning
>was that black and curly-haired are dominant. But, I contend that no matter
>which are dominant, one cannot answer based on the information given.
>
Well, one doesn't need to be a geneticist to answer what is basically a
probability question. At least if the basic principle about dominant and
recessive genes is true... Let's assume that it is.
Your argument is close on the curly hair issue, but incorrect on the black.
Since each parent has at least one black haired gene (just imagine all those
hairy genes:-)), the chances that *one* of the parents will donate a black
haired gene is 1 - .5^2 = .75 so "chances are" that the child will have
black hair. If one of the parents has both dominant genes then the chances
are 100% so infact the chance of black hair is greater than 3/4.
For curly hair, we have only one parent with the trait. If that parent has
one curly gene, then there is a fifty percent chance of the child's hair
being curly. If two then there is a 100% chance. We can't know which but
All we need to assume in order to say that the child is more likely curly
haired is that there is some e>0 which represents the chance that the mother
had a second "curly gene" (Wuck, wuck, wuck! Nyuh!:-)) Since in the "real
world" this is likely, it's not unreasonable to expect that as an answer.
Of course the situation was not elaborated and it's possible that the
mother's father was straight haired, in which case we know that the chances
are 50%... but since we are not told, we must assume that she has the
standard chance of having the extra gene. It doesn't matter what it is, or
that we know it, as long as it is greater than zero.
The only possible objection I have to this is my wonder whether curly hair
is, in fact dominant. I know an awful lot of people with straight hair
(myself included). Keep in mind that lot's of straight haired people get
perms, relatively few curly haired people get their hair straightened...
>Ob. Jeopardy puzzle #1:
>If you get the Daily Double in Double Jeopardy with time running out and
>have $3000, with first place having $11900 and third place having $2500,
>how much should you bet?
obviously, this depends on your confidence in the category. I would bet the
minimum, if I was ancy about the question, figuring I'll go home with as
much consolation money as possible. If I'm confident I bet it all (or 2950)
in order to put myself in a position to win or tie at Final Jeopardy.
>Ob. Jeopardy puzzle #2:
>If you get the Daily Double in Double Jeopardy with time running out and
>have $6000, with first place having $11900 and third place having $2500,
>how much should you bet?
Nothing wouldn't be bad... Especially if you aren't comfortable with the
category. You are already in a position to win or tie in Final Jeopardy.
If I remember correctly, however you must make a minimum bet of $100 or $200
which means that if you miss you can't win/tie unless Mr./Ms. first place
lets you. Therefore, use the same algorithm as in the first question. Only
if you really don't think you'll answer the question, should you conserve.
Figuring out what to bid to not conserve is tough. because any winning bid
will put you in some position to win/tie. figure that if you bet it all,
your chances to win/tie assuming you correctly answer Final, increase from
1/2 to 1. Personally I would bet it all only if I was very confident of the
category. Otherwise I would bet minimum. This also depends on how you
juudge your opponent. Has he just had more answers than you, or is he
simply quicker on the draw? In the first case, he is likely to do better on
final jeopardy than you anyway, in the second, perhaps you will do just as
well, or even better than he, since final jeopardy does not depend nearly as
much on the speed of your answer. This question really does depend on the
situation.
>Ob. Jeopardy puzzle #3:
>Why does Alex seem so surprised when someone wins by "only" $1?
Three possible reasons:
1) He's stupid and doesn't realize that strategy works that way...
2) He's not that stupid but assumes (or his producers assume) that the
audience is, and he says it to play up to the apparent stupidity of the
audience...
3) He's actually expressing surprise that noone's figured out the
work-together-get-ties-and-burn-the-show algorithm... :-)
My money is on #2.
--mike
--
Mic3hael Sullivan, | "Who's paying 20 grand a year to go here,
Society for the Incurably Pompous | and who's calling whom *stupid*??"
University of Rot and Fester | --to someone calling UR admins stupid.
-*-*-*-*-*-*-*-*-*-*-*-*-*-
Dave Seaman
>that I thought had no correct answer (or in Jeopardy terms, an answer that
>had no correct question). It was a genetics question, that, to paraphrase,
>asked what kind of hair the baby of a black, straight-haired father and a
>blonde, curly-haired mother would "most likely" have. The answer given was
>black, curly-haired.
I think I see the assumptions that were used to arrive at the answer.
1. Black hair is dominant; blonde hair is recessive (quite
reasonable).
2. Curly hair is dominant; straight hair is recessive (also quite
reasonable).
3. The black-haired parent has a 50% chance of carrying a recessive gene
for blonde hair (highly questionable assumption).
4. The curly-haired parent has a 50% chance of carrying a recessive gene
for straight hair (also highly questionable).
If you do the arithmetic based on those assumptions, it turns out that the
offspring has a 9/16 chance of having black, curly hair.
--
Dave Seaman
a...@seaman.cc.purdue.edu
John Kemp
>
>>Ob. Jeopardy puzzle #1:
>>If you get the Daily Double in Double Jeopardy with time running out and
>>have $3000, with first place having $11900 and third place having $2500,
>>how much should you bet?
>
>obviously, this depends on your confidence in the category. I would bet the
>minimum, if I was ancy about the question, figuring I'll go home with as
>much consolation money as possible. If I'm confident I bet it all (or 2950)
>in order to put myself in a position to win or tie at Final Jeopardy.
>
consolation money, just prizes. In the old ('60s-'70s, Art Fleming, not-
so-big money) Jeopardy, you took home the money you earned even if you
didn't win the game. Just one of the many changes, few of them good, from
the original show to the new one.
All E-mail replies to below, please. (Operating under a borrowed account and
odd things happen to replies to the net.)
____________________________________________________________________________
Harold Brooks Internet/Bitnet:bro...@uiatma.atmos.uiuc.edu
Dept of Atmospheric Sciences UUCP:{uunet,convex}!uiucuxc!uiatma!brooks
University of Illinois
Larry Baum
: The other night (I think it was 4/2), there was a Daily Double question
: that I thought had no correct answer (or in Jeopardy terms, an answer that
: had no correct question). It was a genetics question, that, to paraphrase,
: asked what kind of hair the baby of a black, straight-haired father and a
: blonde, curly-haired mother would "most likely" have. The answer given was
: black, curly-haired.
:
: I contend that there is no correct answer. I suspect that their reasoning
: was that black and curly-haired are dominant. But, I contend that no matter
: which are dominant, one cannot answer based on the information given.
: We can know the genetic make-up of a parent that has a recessive trait
: (both genes for that trait are recessive), but we cannot know the genetic
: make-up of the parent with the dominant gene (we know one of the two is
: dominant, but we do not know the other). Therefore, the parent with the
: dominant trait does not necessarily pass on the dominant trait to the
: baby, so we cannot know what trait the baby will have.
Let the hair-color genes for P1 be c1 and c2. c1 is black and c2 is
unknown,
Let the hair-color genes for P2 be d1 and d2. Both d1 and d2 are blond.
(I don't know how red-headedness works, so I ignore it.)
There are four cases:
c1d1: black
c1d2: black
c2d1: ???
c2d2: ???
The last two cases will be black unless c2 is blonde.
So the probability that the baby's hair is black is 1/2 + 1/2P where P
is the probability that c2 is black. Certainly P > 0, so the overall
probability is > 1/2.
:
: Ob. Jeopardy puzzle #1:
: If you get the Daily Double in Double Jeopardy with time running out and
: have $3000, with first place having $11900 and third place having $2500,
: how much should you bet?
:
$3000. Second place is relatively worthless and unless you have more
than 1/2 of player 1, he/she will win by default.
: Ob. Jeopardy puzzle #2:
: If you get the Daily Double in Double Jeopardy with time running out and
: have $6000, with first place having $11900 and third place having $2500,
: how much should you bet?
This is more tricky. It depends on your confidence regading the DD.
According to Games magazine, the difficulty of the DD corresponds to its
original $ value (e.g. if it is in a $200 square, it should be
relatively easy; if it is in a $1000 square, you should watch out.)
If you are confident of success, bet the $6000. If you miss, you're
dead anyway, so you might as well go for broke.
If you are not confident, bet nothing (can you do that?). You don't
wan't to go into FJ with less than 1/2 player 1.
:
: Ob. Jeopardy puzzle #3:
: Why does Alex seem so surprised when someone wins by "only" $1?
:
It's his job to make the viewer think the show is exciting.
--
Larry Baum
Advanced Technology Center
Boeing Computer Services uucp: uw-beaver!bcsaic!lbaum
(206) 865-3365 internet: lb...@atc.boeing.com
Larry Baum
But black hair and curly hair may be highly correlated; i.e. these may
not be independent variables.