"We know two aspect to these partial results, one thorough observation and the other through deductive reasoning. Eliminate the impossible from the experiment."

Absolutely, but don't change what the experiment was. That is why you must toss two coins and then just take the cases that give at least one boy. That is the universe we're in. You run the experiment under consideration, eliminate those that are inconsistent with your situation (GG), and then look at the rest.

@uclabb - You never asked a question anywhere in that post.

" ""If you know only that (a) two coins were tossed and (b) at least one was heads, there is a 50% chance that both coins were heads." "

Yes! Yes! Yes!"

In the other thread, I wrote this:

"a) Draug says that, if two fair coins are tossed and that afterward, you learn that heads came up on at least one, there is a 50% chance that they both came up heads;"

You said, "I disagree with a."

Confused much?

Draug- What you just "quoted" isn't in what I freaking copied and pasted for you for easy access.

Okay, respond to my request, then.

"Please either explain why you think it is logically unsound to pick the BB/BG/GB/GG combination first, or why (without reframing) there is any justification for doubling the weight of the third case."

OK, to clarify, I disagree with A for designing the experiment because the rules of our experiment do not allow for two fair coin tosses. TT can never happen but it will on two fair coin tosses repeated over and over and that affects the outcome.

Interesting as I found all this I don't see how we can get past the fundamental disagreement that we know one is a boy so lets now look at the other ( i even provided a very simple step by step process for you), and your viewpoint that we can't discount the possibilities of no boys (even though that's a condition).

Really I think your experiment is akin to a science experiment where you fail to properly ensure the control variable, is, in fact, controlled.

Exactly, Draug. So you agree with both of the things that I said. The first one was what I said you assert. The second was how I said you assert it should be tested.

So far as I can tell, you agree with both A and B.

OK, my response to uclabb.

We have a known constant K that will always be a B. This eliminates GG. This known constant K can either be child 1 (order it however your wish) or child 2.

If child 1, we have KB and KG as possibilities.
If child 2 we have BK and BG as possibilities.

That means we have KB, BK, KG, and GK as possibilities. Seeing as K has feature B, BB appears twice and therefore gets weighted twice when computing the probability.

So here's a question, Draug.

Given that we're talking about THE RESULT OF TWO FAIR COIN TOSSES, can you think of ANY way to test your claim that actually involves the random simulation of TWO fair coin tosses?

Yes, so design an experiment that eliminates TT form a possible result and run it. I can write the code if yo like. VB or VC#, your choice (prefer C# as I live and breath in it)

Again, we aren't truly talking about tow fair coin tosses. We are talking about two fair coin tosses in which one is guaranteed to be Heads and double tails can never happen. That is the rule of the probability we seek. allowing double tails invalidates the results just as allowing O2 into the chamber where we are testing bacterial growth invalidates those results.

"If child 2 we have BK and BG as possibilities.
"

Correction that should have been

If child 2 we have BK and GK as possibilities.

Draug, it's obvious though that changing the method of generating the data changes the probability. Let me give you a stupid example.

If I throw two die, and know that one of them is coming up 1, what is the probability that the other did?

If I throw two ten-sided die, and know that one of them is coming up 1, what is the probability that the other did?

Irrespective of our disagreement about whether the answer to the first is 1/6 or not, we can both agree these questions will have different answers.

The question at issue here is about odds from data that were generated in a very particular way (two fair coins). Any simulation that ignores that will be completely invalid.

You are systematically confusing the generation of results with knowledge about the results of that generation. Just because we know that at least one head happened to come up does not justify our ignoring the fact that it came up on two random coin tosses.

We must precisely toss two coins and then keep only those that are consistent with our information. That is how you do probability. You look at all the space of results generated BY THE PROCESS YOU ARE TALKING ABOUT, and you throw away those that are inconsistent with what you know, and you use the rest to compute probability.

Because you are ignoring this, and seemingly asserting that you can just ignore where the data are coming from, I don't know what else to say.

Except to ask, one more time, if you can come up with ANY experiment that actually involves tossing TWO coins. If you cannot -- yet your claim is about the result of two fair coin tosses -- then that is extremely telling.

"Again, we aren't truly talking about tow fair coin tosses. We are talking about two fair coin tosses in which one is guaranteed to be Heads and double tails can never happen."

No, that is false. We're talking about two truly fair coin tosses where, AFTER THE FACT, we learn that at least one did come up heads.

I have been crystal clear about the situation I am talking about, repeating it ad nauseum, so there is no excuse for saying there has been lack of clarity.

No, probability is not calculated that way when you put constraints on what can happen. We have a constraint that one must be a male or one must be heads. If you are going to ignore that constraint in you random factor to prove the probability, then you may as well make the coins have 3 sides or the kids have 3 genders. After all, one of the kids could be truly transsexual.

We have two constraints in computing this probability.
A) that the objects have a binary state we are computing the probability for.
B) that at least one of the objects is already set at a particular state (there fore no coin toss for that said object, just a random selection as to *which* object is that object).

"No, probability is not calculated that way when you put constraints on what can happen."

There is no constraint on what can happen. There is a constraint on what did happen. I'll paste again for the nth time.

"You know only that (a) two coins were tossed and (b) at least one was heads."

There is no constraint on what can happen. Notice the past tense, "was heads"? It's after the fact. You are insistently ignoring the first part, that two coins were tossed.

But they were. That's what gave the heads. I am not ignoring the constraint. That is why I wipe out the GG from the probability space. It's you who are ignoring. It's you who continue to make a claim about the result of two coin tosses but can't write a simulation that actually includes two coin tosses.

Take care.

"B) that at least one of the objects is already set at a particular state (there fore no coin toss for that said object, just a random selection as to *which* object is that object)."

This is where you are confused. One of the objects is not set at a particular state. It is set that one of the objects (but not a specific one) ended up in a particular state. Do you not see the difference?

There is no difference because we randomly choose which object has that state. Do you not see that this eliminates the impossible and only leaves the possible for computing?

Module Module1

Sub Main()
Randomize()

Dim bCoinAHeads As Boolean
Dim bCoinBHeads As Boolean
Dim iToss As Integer
Dim iCoinToToss As Integer
Dim lDoubleHeads As Long = 0
Dim lSplit As Long = 0
Dim dOdds As Double


For lLong = 0 To 100

iCoinToToss = CInt(Math.Ceiling(Rnd() * 2))
If (iCoinToToss = 1) Then
'Toss Coin A
iToss = CInt(Math.Ceiling(Rnd() * 2))
If iToss = 1 Then
'Coin A is heads
bCoinAHeads = True
Else
'Coin A is tails
bCoinAHeads = False
End If

'Assign Coin B as Heads
bCoinBHeads = True
Else
'Assing Coin A Heads
bCoinAHeads = True

'Toss Coin B
iToss = CInt(Math.Ceiling(Rnd() * 2))
If iToss = 1 Then
'Coin B is heads
bCoinBHeads = True
Else
'Coin A is tails
bCoinBHeads = False
End If
End If

If bCoinAHeads = bCoinBHeads Then
' Add one to double heads count
lDoubleHeads = lDoubleHeads + 1
Else
'Add one to split count
lSplit = lSplit + 1
End If
Next

dOdds = lSplit / lDoubleHeads


Console.WriteLine("Ratio (Split to Doubles)=" + dOdds.ToString())

Console.ReadKey()

End Sub

End Module

@smeck and uclab, did you read the pages in between last night and when you started posting? I simplified the problem to its bare essentials and even did the math to show that it works too given that you make the proper assumptions. You took it back to the necessary complex, after I had spent so much effort simplifying it to the essence of the problem. I explained why the double BB was invalid in a very simple thought experiment. The discussion seems to have regressed in the time it took for me to cook dinner.

another way to look at it that truly encompasses all the possibilities that we count. we have three criteria, known boy, boy, girl. given this is the outcome it should be fair to say those are the possibilities for the first child. the known boy can produce an unkown boy or an unknown girl for the second child, the boy and girl can only produce a known boy as their outcome, giving us 2 known boys with 2 unkown boys, and 2 known boys with 2 unknown girls, and a fifty fifty chance.

Run that. I got the following output:

Ratio (Split to Doubles)=1.02

necessary complex = unnecessary complex.

Making a choice before an experiment and making an observation after an experiment is fundamentally different. I don't know how to continue if you do not see that.

Actually, Draug, let me ask you a somewhat different question. (For this question, ignore issues such as transgender, different birth ratios, etc. -- assume 50/50 birth ratios again)

Consider the set of all families in the US who have two children, at least one of whom is a boy.

What percentage of those families have two boys?

And draug, if you live and breath c# why didn't you write it in c#? What's this VB nonsense? :-P

Sorry philcore, I must have missed. I read some of that, not all. Do you have a particular post in mind, or page number? I'd be happy to go read it.