Poll of the Day > Help me understand your reasoning in solving the following easy problem

(+) Yeah! (+)

SKARDAVNELNATE posted...

Wait, what? G1 is the one that the question specifies is a girl. The bold ones make no sense.

Neither does the way you laid it out. That was kind of my point.

If you know that at least one of the children is a girl, but you don't know which one, there are only three possibilities:
-Child A is a girl and Child B is a boy
-Child A is a boy and Child B is a girl
-Child A is a girl and Child B is a girl

There's no way you can rearrange that to get another option and no way to reduce that to remove one of the options. Similarly, saying "Child B is a Girl and Child A is a Girl" - which you were trying to do with your "G1/G2" notation - is exactly the same as the third option; it's not a separate and distinct outcome.
---

SKARDAVNELNATE posted...

2) To sync up with the original question one of the coins has already been flipped for us. We are only flipping the remaining coin. Why then would one side inherit a greater probability than the other?

Because that's an incorrect reading of the question - one that is quite common and that trips people up when they try and figure out the probability for this exercise.

Yes, if you flipped one coin and it turned up heads then flipped another coin, the odds of it also being heads are 50%. The fact that the first coin was heads has no impact on the probability of the second toss also coming up heads.

But that's not the situation we have here. The situation here is that I flipped two coins behind I a screen and I tell you that at least one of them came up heads. In this case, you don't know whether it was the first coin I flipped, the second coin, or both of them - all of those scenarios would satisfy the condition I gave you. In that case, there's a 33% chance that just the first coin is heads, a 33% chance that just the second coin is heads, and a 33% chance that both coins came up heads.

In essence, the fact that you have no way of knowing which coin I'm talking about when I say "At least one of them came up heads" is what drives the probability, because you can't think of it as a sequence; you have to think of it as a pairing when presented with the information that way, because I could be talking about either coin (or both of them).

We can relate this back to the original problem thusly:

If I told you I had two kids and suddenly one of them ran up to me and you saw she was a girl, the odds of my second child also being a girl are 50%. This is because you know which child is the girl - the one standing in front of you - meaning the child you don't see has even chances of being a boy or a girl.

But that's not the question as given. The question as given would be if I told you that at least one of my two kids is a girl. You have no way of knowing "which" kid of mine is a girl, meaning there's a 33% chance just the older child is a girl, a 33% chance just the younger child is a girl, and a 33% chance both of them are girls.

It's weird, but the math checks out. Again, for this problem, you can't think "I know something about X, I just need to know about Y" or you'll get the wrong answer; you have to think "I know something about X/Y as a pairing, I need to find out something else about that pairing".
---

darkknight109 posted...

If I told you I had two kids and suddenly one of them ran up to me and you saw she was a girl, the odds of my second child also being a girl are 50%. This is because you know which child is the girl - the one standing in front of you - meaning the child you don't see has even chances of being a boy or a girl.

But that's not the question as given. The question as given would be if I told you that at least one of my two kids is a girl. You have no way of knowing "which" kid of mine is a girl, meaning there's a 33% chance just the older child is a girl, a 33% chance just the younger child is a girl, and a 33% chance both of them are girls.

It's weird, but the math checks out.

That is super weird! It has to do with the "luck" that just the single girl would run up to you if you had a Boy-Girl pair, right? Since a girl ran up to you, the odds of having two girls increase a little.
---

darkknight109 posted...

-Child A is a girl and Child B is a boy
-Child A is a boy and Child B is a girl
-Child A is a girl and Child B is a girl

There's no way you can rearrange that to get another option and no way to reduce that to remove one of the options.

And that is what I find inconsistent.
Set 1) A is the one we were told is a girl.
Set 2) B is the one we were told is a girl.
Set 3) Either A or B could be the one we were told is a girl.

Sets 1 and 2 are only different because the known girl is A or the known girl is B. If that is enough to differentiate them, then set 3 should be split into 2 possibilities.

Set 3 has one outcome, 2 girls. Sets 1 and 2 share an outcome, 1 of each. If the outcome of set 3 means it's irrelevant which of A or B was previously known, then it should also be irrelevant for sets 1 and 2.
---

darkknight109 posted...

Yes, if you flipped one coin and it turned up heads then flipped another coin, the odds of it also being heads are 50%. The fact that the first coin was heads has no impact on the probability of the second toss also coming up heads.

But that's not the situation we have here. The situation here is that I flipped two coins behind I a screen and I tell you that at least one of them came up heads.

The result of one coin flip doesn't influence the other just as whether I flipped a coin or you flipped a coin should have no influence on the result.
---

SKARDAVNELNATE posted...

darkknight109 posted...

Yes, if you flipped one coin and it turned up heads then flipped another coin, the odds of it also being heads are 50%. The fact that the first coin was heads has no impact on the probability of the second toss also coming up heads.

But that's not the situation we have here. The situation here is that I flipped two coins behind I a screen and I tell you that at least one of them came up heads.

The result of one coin flip doesn't influence the other just as whether I flipped a coin or you flipped a coin should have no influence on the result.

Imagine the following: I flip 100 coins behind a screen, I look at the results and cover up a single one of the coins. I then remove the screen and show you the result. You see that every one of the 99 coins you can see are heads. I then ask you to guess which side is up on the coin I'm covering up. What should you answer? Is it really 50-50? The trick is to realize I could cover up ANY of the 100 coins. It could be the first coin I flipped, the second coin, the thirty-first coin, etc. Since EACH of the 100 coins would have to come up heads for the coin I'm covering up to be heads, but only a single one of the 100 coins would have to be tails for it to be tails, the probability becomes a lot bigger that it's tails!

Do you follow?
---

DarknessLink7 posted...

Do you follow?

You stated previously, "The fact that the first coin was heads has no impact on the probability of the second toss also coming up heads." The other 99 coins don't influence the remaining coin. Any one coin in the 100 has the same odds of heads as any other coin.
---

SKARDAVNELNATE posted...

DarknessLink7 posted...

Do you follow?

You stated previously, "The fact that the first coin was heads has no impact on the probability of the second toss also coming up heads." The other 99 coins don't influence the remaining coin. Any one coin in the 100 has the same odds of heads as any other coin.

Yes. The fact that the first coin was heads has no impact on the probability of the second toss also coming up heads. That is true. But do you follow the example I posted?

Since every single one of the 100 coins would have to come up heads for the single coin I'm covering up to be heads, the probability for that is really low. But on the other hand, only a single one of the 100 coins would have to have come up tails for the single coin I'm covering up to be tails. The probability of getting at least one tails if I flip 100 coins is very high. Very very high.
---

DarknessLink7 posted...

But do you follow the example I posted?

I follow your example. I don't agree with your conclusion.

We're not dealing with some large table of coins. We know the 99 are heads. That leaves (99H, 1T) or (99H, 1H) as the remaining possibilities. Unless somehow in performing the ritual the probability which didn't occur was transferred to certain coins but not others. And at this point we're so deep in hypotheticals that we are talking about a magic ritual.

You have presented multible ways of rewording the same example and I keep circling the same practical reasoning as to why your example explains nothing. I don't think you understand why a coin flip works out that way any better than I do.
---

SKARDAVNELNATE posted...

DarknessLink7 posted...

But do you follow the example I posted?

I follow your example. I don't agree with your conclusion.

We're not dealing with some large table of coins. We know the 99 are heads. That leaves (99H, 1T) or (99H, 1H) as the remaining possibilities. Unless somehow in performing the ritual the probability which didn't occur was transferred to certain coins but not others. And at this point we're so deep in hypotheticals that we are talking about a magic ritual.

You have presented multible ways of rewording the same example and I keep circling the same practical reasoning as to why your example explains nothing. I don't think you understand why a coin flip works out that way any better than I do.

Maybe I don't understand it. It's hard to know with these things. But please bear with me for a while longer. I have something interesting I want to show.

If you accept that the probability of all 100 coins coming up heads is very very low, and you accept that the probability that one of them came up tails is a lot higher, then imagine this:

I tell you that the coin I'm covering up is the seventeenth coin I flipped.

Now suddenly the odds of it being tails is back to 50%. If you don't know which coin I'm covering up, the odds of it being tails is a lot higher, but the moment I tell you which coin it is, it goes back to being 50-50.

This example shows that knowing the order of events affects the probability of other events.
---

DarknessLink7 posted...

That is super weird! It has to do with the "luck" that just the single girl would run up to you if you had a Boy-Girl pair, right? Since a girl ran up to you, the odds of having two girls increase a little.

More to do with the fact that you can now tie the information you were given ("At least one girl") to an identifiable part of the pairing ("The first child you met" or "The child you can see"). This changes the question from "A or B is a girl, what are the odds they're both girls?" to "A is a girl, what are the odds that B is also a girl"?

SKARDAVNELNATE posted...

If that is enough to differentiate them, then set 3 should be split into 2 possibilities.

But look at the set-up - there's no way to get more possible outcomes from it. Again:

-Child A is a girl and Child B is a boy
-Child A is a boy and Child B is a girl
-Child A is a girl and Child B is a girl

If you try and get another possibility out of the third option, you get "Child B is a girl and Child A" is a girl, which is identical to what we've already listed as the third option.

SKARDAVNELNATE posted...

If the outcome of set 3 means it's irrelevant which of A or B was previously known, then it should also be irrelevant for sets 1 and 2.

I don't follow your logic here.

It doesn't matter which of the kids was "previously known"; that still doesn't mean the first two sets aren't different outcomes. Swap out "Child A and Child B" with "The older child and the younger child" if it helps you visualize it better.
---

SKARDAVNELNATE posted...

We're not dealing with some large table of coins. We know the 99 are heads. That leaves (99H, 1T) or (99H, 1H) as the remaining possibilities. Unless somehow in performing the ritual the probability which didn't occur was transferred to certain coins but not others. And at this point we're so deep in hypotheticals that we are talking about a magic ritual.

Not magic at all - hell, you can do this experiment for yourself. All it takes is a bunch of coins and a screen.

Consider the problem via the following scenarios:

1) You flip 100 coins. What are the odds you get at least one tails result?

You don't need to actually run the math, but you know it's pretty high, right? There's a (significantly) greater than 99% chance that at least one of your coins is going to come up with a tails result. This is because in order for you not to get a single tails result, every single coin must come up with heads - the odds of that are 0.5^100 (or 7.9 x 10^-29 percent) - pretty remote.

2) You flip a single coin. What are the odds you get exactly one tails result?

50%, right? Easy. But now let's change it a little:

3) You flip 100 coins. What are the odds you get exactly one tails result?

Again, you don't need to run the math here, but you should also instinctively know that the result is not 50%. Even though I'm only asking you about a single coin toss - the same way I was in the last question - the fact that it's part of a set changes the odds. Now you don't just have to flip a single tails result, you have to flip a tails and 99 heads, which substantially reduces the odds.

4) You flip 100 coins. What are the odds you get 99 heads, followed by tails on the 100th flip?

Same odds as tossing all heads, right? Because I'm setting the results of all 100 flips.

5) You flip 100 coins. What are the odds you get exactly one tails result, regardless of when it occurs?

This is the same question as Question 3. But does it have the same answer as Question 4? And if not, is it less or more likely?

The answer is not the same as Question 4 - even though I'm specifying the results of all 100 flips, I'm not setting a required order the way I am in Question 4. It's easier to get a single Tails result if you don't specify on which flip it has to occur, because then you have 100 different possible "successful" results, whereas if you specify it has to occur on, say, the 100th flip, you only have one possible "successful" result.

6) If I flip 100 coins in secret, then tell you "I got 99 heads on the first 99 flips," what are the odds the 100th flip is also heads?

The answer is 50%. The fact that the first 99 tosses were heads has no impact on the 100th flip.

7) If I flip 100 coins in secret, then tell you "I got at least 99 heads", what are the odds I got all 100 flips heads?

The answer here is NOT 50%. Again, refer back to Questions 4 and 5. Because you don't know which flips are the 99 I was talking about, there are 101 different scenarios that would satisfy the information I gave you (a single tails flip on any of the 100 flips, plus the result of all 100 flips being heads) and only one of those is an "all heads" result, meaning my odds of having 100 heads is substantially lower than having 99 heads and 1 tails.



If you follow this through, you should be able to see that information in sequence (I.E. Child A is a girl, what are the odds that Child B is also a girl) produces different results than information about sets (I.E. at least one of Child A and B is a girl; what are the odds both are girls). You're thinking about the problem as the former, but it's actually a question of the latter, which is why you're not getting the result you should.
---

I just noticed that that the people I've been debating with are both Dark-something. I may have confused your identities at some point. However, it seems that you both believe the result of one coin does impact on the probability of the other coins. In which case I picked up the wrong book earlier. I should have looked into quantum entanglement.
---

SKARDAVNELNATE posted...

I just noticed that that the people I've been debating with are both Dark-something. I may have confused your identities at some point. However, it seems that you both believe the result of one coin does impact on the probability of the other coins. In which case I picked up the wrong book earlier. I should have looked into quantum entanglement.

Except, if you'll note, I've explicitly mentioned multiple times that's not the case.

It's not that "the result of coin flip X changes the result of coin flip Y", it's that "you know the result of a coin flip, but are uncertain as to whether that result applies to X or Y". That uncertainty is what's affecting the probability.
---

SKARDAVNELNATE posted...

it seems that you both believe the result of one coin does impact on the probability of the other coins

past coin flips do not affect future coin flips, but partial information does impact what we can say we know about past coin flips. otherwise, we'd have to say that the chance is actually only 0% or 100%; it either happened or it didn't.
---

Bump!

I'm not ready to have this deleted yet, sorry. I want to actually take some time to go through it and take notes of what you all said first.
---

THen I will bump it for you.
---