On probability theory and the two children (or two coin) problems
- From: "Beldin the Sorcerer" <beldinyyz@xxxxxxxxxxx>
- Date: Sun, 03 Aug 2008 01:05:30 GMT
Hi, I'm the actor who plays Beldin the Sorcerer on the internet. I'll be
answering all of Wuzzy's posts in full character quite shortly, but before
then, in case anyone who actually doesn't understand probability is
following the various threads, I thought I'd take a few minutes to explain
the actual math and science.
What is "Probability"?
Largely, it's an estimate based on known quantities of something that will
happen in the future.
It is a number between 0 (never happen) and one (certainty)
What is the Probability of a past event?
When we talk about that, what we really mean is, what is the statistical
likelihood of something, given the information that we know. The actual
"probability" of something that must have 'already happened' to be true is
of course either 0 or 1. Given a large number of cases, the statistical
likelihood is, X% will be true, and it is common to use the word
'probability' in discussing that.
How does information change the probability?
It eliminates some possible cases. It also MAY make some of them less likely
than others.
Example : I deal a deck of cards into 4 equal piles. What's the probability
of the ace of spades being in any given pile? 25%
I show you three cards from the first pile and they aren't the ace of
spades. I look at the entire fourth pile without showing you and tell you
truthfully "It's not here" . The probability of it being in the fourth pile
is now 0. The probability of it being in the second or third pile is higher
than being in the first pile, because you've eliminated some of the cards,
but not all of the cards.
"How does this relate to the sibling problems"
These are commonly given as introductory problems in a probability course.
Indeed, they can, and often do, lead to heavy argument, and frequently the
teacher can be wrong, if they weren't paying enough attention when writing
the problem.
There are two basic kinds of information "Omnicient information about pairs"
and "Specific information about individuals".
It isn't always obvious which is which. You need to think about the issue a
bit before deciding.
Example. A woman says "I have at least one son". Not a common, everyday
exclamation, is it. It's contrived for the sake of the problem. This is
omnicient information.
A woman says "I have a son, Dirk. He's on the honor roll". This is specific
information.
What's the difference?
Well, these kinds of probability puzzles are often about determining the
likelihood of some combination of gender. Some deal with Pairs of kids, some
deal with Partners.
What's the difference?
Consider the following cases. Four idealized sibling pairs :
B b
B g
G b
G g
There are four 'pairs' and eight 'partners'. Eliminating information must be
used correctly to determine what exactly you're looking at.
Consider twenty 2 children families, 5 each of the above. They and their
parents come to a halloween party. The kids are in full costume, so gender
isn't obvious. The parents are in a different room, and you're talking to
them.
"How many have at least one boy?'
15 sets of parents raise their hands.(75%) This is an omnicient information
question.
Of you, how many have a girl too? 5 sets of hands go down, 2/3 have a girl.
That's the probability of a daughter. 1/3 would be the answer for the
probability of a second son.
You go to the costumed kids. "How many of you have a brother?"
Twenty kids (50% of them) raise their hands. Half the kids are boys, that
should be obvious.
When it starts to get confusing is when people get clever about concealing
the nature of the information, and perhaps screw up.
Consider the parents again. A parent, at random, shows you a picture of
their son. This is specific information, and it means there's a 50-50 chance
for the gender of their sibling.
You ask the parent "do you have at least one boy" and they say yes, and show
you his picture. This is NOT the same thing. This is now omnicient
information.
Wuzzy would go on at length here about how the parent "knows" the same thing
in both cases. That's irrelevent.
Go back to the 20 sets of parents. 15 raise their hands to say "Yes, we have
at least one son". Once that's established for certain, they can all show a
picture of a son. 10 sets of parents have no choice, 5 sets of parents still
need to choose which son to show you. This is the key, of course. If that
choice is made PRIOR to you learning about the gender, it needs to be
factored in. If it is made AFTERWARDS, it doesn't change the likelihood of
you getting more information, it just makes it less useful.
It's not about whether the parent "knows" both genders. Presumably, they all
do. It's about whether they made a choice yet.
Someone else knowing whether the specific case is true or false doesn't
change the statistical likelihood (and therefore what we generally call
'probability' at all UNLESS they give you information relevent to it.
Shuffle a deck of cards. What's the probability of the top card being the
ace of spades? 1/52. If I look at it, and say nothing, from my standpoint
it's either 1 or 0 now. From your standpoint, it hasn't changed.
Flip two coins without telling me what they are. Look at them both. Now tell
me one coin. Telling me it's Heads, or Tails, isn't omnicient information,
even though you know both coins. Not because you don't know it, but because
prior to the question, I didn't KNOW you knew it. If you have a matched pair
(HH or TT) you will always give me one answer. If you don't, you'll always
have a choice, so that choice needs to be discounted by the likelihood of
you MAKING that choice.
Wuzzy doesn't get this. He may never get this. But YOU need to get this to
understand probability.
.
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