Poll: A little math problem

[Samirat]

Yeah, we know one is heads. BUT WE DON'T KNOW WHICH ONE.

Sure we do--the one that isn't known.

Take a look at this variation on the Three Card Problem:

http://www.escapistmagazine.com/forums/jump/18.73797.825805

So I went ahead and wrote the program, here's the output of 5 runs of 100000 flips each:

We're not asking about 5 runs of 100000 flips each, we're talking about one run of one flip: you've fallen for the Gambler's fallacy. [http://en.wikipedia.org/wiki/Gambler%27s_fallacy]

That's not the gambler's fallacy at all. You post the wikipedia page, but you don't bother to read it. Gambler's fallacy is where, after flipping one coin and seeing it land heads, you assume that the next coin you flip is more likely to land tails.

Problems like this and Monty Hall play on nonexamples of the Gambler's fallacy, making you believe that two events are independent, and are therefore equally probable, when in reality they aren't. For instance, Monty Hall makes you think that out of the two doors, there is an equal possibility that it could lie behind either. This makes you think that because at least one of two dogs is male, there is an equal possibility that the other could be male or female. Both use the opposite of the gambler's philosophy, making people believe that they are victims of the fallacy when they really aren't.

 

[Samirat]

You flip 4 coins and ask your friend if two of them are heads. He says "yes." What are the chances that the other two are heads compared to the chances that the other two are comprised of 1 head and 1 tails.

By your logic:
KKHH, KHKH, KHHK, HKKH, HKHK, HHKK = 6 possibilities

KKHT, KHKT, HKKT = 3 possibilities

I'm sorry but, it's not working out between us. You just don't get me, respond to quickly, or something, but we just can't get through to each other.

I mean, if you think my logic would be:

KKHT, KHKT, HKKT

and not something more like (just doing this fast now without double checking)

KKHT, KHKT, HKKT, HKTK, HTKK, KHTK, KTHK, HTKK, THKK, KKTH

You either don't understand me, how to do permutations, or both.

Sorry I couldn't be of more help.

Is it just me, or does that look like, I don't know, a listing of Soviet agencies?

Yeah, you're right, there should be 12 of the things. 4 spaces for the T, times 3 arrangements for the other three. Which still isn't correct. The chances are

6 2 heads 2 tails
4 3 heads 1 tails
1 4 heads

I really do want you to justify this to me, Cheeze.

 

[Alex_P]

Okay, one big fundamental misconception I keep seeing referenced over and over but never directly addressed. It sounds kinda like this:

"Two females don't satisfy the 'at least one male' requirement, so any experiment that has the potential to produce the two females result is an inappropriate model for this scenario."

That's pretty much gambler's fallacy at work. You're saying "This fair coin was heads this time so it can never be tails!"

Untrue! It's still a fair coin.

And, yes, that means that if you simulate the problem with two fair coins, then the answer to "Is at least one male?" is gonna be "no" 25% of the time. That's how it works. That's okay.

If you do something like setting one coin to make sure that you can never get TT, then what you're actually doing is breaking the "fair coin" rule completely -- your model ceases to be representative of the problem.

-- Alex

 

[Alex_P]

Funny because to me, that's what everyone else is saying--'watch me use the Law of Large Numbers to come to a conclusion about...the smallest possible number!'

Simply using a large number isn't an application of the law of large numbers. The law of large numbers has to do with correlation between an expected average (from a probability calculation) and the actual observed value.

Not to mention that fact that the people who claim to have the right answer can't agree on the right method, which is...kinda troubling, don't you think?

*shrug* Discrete math isn't something that's taught effective in high school. Most people pick it up haphazardly.

Try this out, and see what you think:

Let's say RED represents a male puppy, and WHITE represents a female puppy. So a card with two RED sides represents a male pair, a card with one RED side and one WHITE side represents a mixed pair, and a card with two WHITE sides represents a female pair.

When we learn there is at least one male, that information allows us to remove the card with two WHITE sides, right? So our 'pool' is one card that is RED on both sides, and one card that is RED on one side, and WHITE on the other. What are the chances of picking either card? 50/50, right? So what are the chances of a male pair vs. a mixed pair? 50/50, just like the cards that represent those two possibilities.

That sound right to you? I think even if it doesn't it gets to the heart of this disagreement.

If you have one card of each type, that's correct. (However, one card of each type is an incorrect model for this particular.)

Now, what if I have X red cards, Y mixed cards, and Z white cards? What are the individual probabilities? How do they change when I take all the white cards away?

-- Alex

 

[Blind0bserver]

I said there was a 50% chance. After all, in a female... well, anything, she has two X chromosomes with the potential of one of them being copied to her offspring. Either way the female is passing on an X. The male has both an X and a Y chromosome, meaning that there's a 1 in 2 chance that he passes on the opposite chromosome that the mate did.

So it's a 50-50 chance.

 

[Alex_P]

Why are we simulating with two fair coins? We don't have two fair puppies!

Actually, I think I was wrong about insisting on throwing out the FFs. Then again, I don't think it matters if you throw out any FFs anymore, either. But I'm not certain about that.

I just still don't understand why we're using two fair coins to simulate one fair puppy and one unfair puppy.

What makes one puppy "unfair"?

-- Alex

 

[Samirat]

Nope - the actual probability of getting one head and one tail is twice that of getting 2 heads.

EDIT: Not when you're only flipping one coin--see the original problem: we know one of the 'coins' is heads. We only ever flip one coin--the value of the non-flipped coin is always Heads/Male.

Try it - flip a pair of coins 100 times. You'll end up with approx. 25 H-H pairs, 25 T-T pairs, and 50 mixed pairs.

Heck, I could write you a program that would do thousands of tests if you'd like, it's pretty simple...

Think about it--if the odds are the same after one additional trial as after 100, then there's something seriously wrong:

"It is important to remember that the LLN only applies (as the name indicates) when a large number of observations are considered. There is no principle that a small number of observations will converge to the expected value or that a streak of one value will immediately be "balanced" by the others. See the Gambler's fallacy." [http://en.wikipedia.org/wiki/Law_of_large_numbers]

Probability does have an impact upon small numbers of observations, however. It determines how likely a certain outcome is. You can't defend your probability based on an argument that probability actually isn't valid on a small scale. Even if this were actually true.

And again, this isn't the Gambler's Fallacy.

 

[Samirat]

Okay, one big fundamental misconception I keep seeing referenced over and over but never directly addressed. It sounds kinda like this:

"Two females don't satisfy the 'at least one male' requirement, so any experiment that has the potential to produce the two females result is an inappropriate model for this scenario."

That's pretty much gambler's fallacy at work. You're saying "This fair coin was heads this time so it can never be tails!"

Untrue! It's still a fair coin.

And, yes, that means that if you simulate the problem with two fair coins, then the answer to "Is at least one male?" is gonna be "no" 25% of the time. That's how it works. That's okay.

If you do something like setting one coin to make sure that you can never get TT, then what you're actually doing is breaking the "fair coin" rule completely -- your model ceases to be representative of the problem.

Why are we simulating with two fair coins? We don't have two fair puppies!

Actually, I think I was wrong about insisting on throwing out the FFs. Then again, I don't think it matters if you throw out any FFs anymore, either. But I'm not certain about that.

I just still don't understand why we're using two fair coins to simulate one fair puppy and one unfair puppy.

Neither of them is an unfair puppy. If you have two coins, and one of them is heads on a particular flip, the coin is still fair.

Now that you've released the primary flaw in your argument, that any experiment where FF's appear is invalid, it should be pretty easy from here. Just perform it.

Two coins. At least one heads. If there are two tails, it doesn't matter, reflip. This probability should be correct for ALL pairs where at least one coin is heads. Therefore, if you have, out of 200 flips around:

50 tails tails, 100 1 heads, 1 tails, and 50 heads heads (this is the most likely distribution)

About 150 of these cases should have the same probability as in out problem. So if you use 50 percent, you end up with 75 pairs of heads heads, which is, of course, incorrect. Only 33 percent will return you the correct distribution here.

 

[Alex_P]

What makes one puppy "unfair"?

The fact that we know that it is male.

No. You're conflating the general probability distribution with an observed fact about this one particular instance.

That's a "gambler's fallacy" (sometimes called "inverse gambler's fallacy").

-- Alex

 

[Alex_P]

No no no!

Where did the two puppies in the problem come from? How were they initially selected?

-- Alex

 

[Samirat]

Why are we simulating with two fair coins? We don't have two fair puppies!

Actually, I think I was wrong about insisting on throwing out the FFs. Then again, I don't think it matters if you throw out any FFs anymore, either. But I'm not certain about that.

I just still don't understand why we're using two fair coins to simulate one fair puppy and one unfair puppy.

What makes one puppy "unfair"?

The fact that we know that it is male.

It doesn't affect each individual puppy. It affects both together, as a pair.

Each individual puppy still has a certain chance of being male, and a certain chance of being female. Neither is guaranteed to be male 100 percent of the time.

 

[Samirat]

What makes one puppy "unfair"?

The fact that we know that it is male.

No. You're conflating the general probability distribution with an observed fact about this one particular instance.

That's "gambler's fallacy."

We're not looking to figure out general probability distribution, we're trying to figure out something about this one particular instance.

Like you said, That's "gambler's fallacy."

Gambler's fallacy is where you assume something about an event based on outcomes already observed. Like when you flip a coin, see that it's heads, and expect the second one to be more likely to be female. That's gambler's fallacy, it's quite simple.

I don't think either of you are correct when you fault it to the other man.

Also, one particular instance has the same probabilities as a million identical ones. The size of your experiment is irrelevant, in terms of probabilities. The size of your experiment just determines how accurate the results will be, in terms of outcomes.

 

[FrcknFrckn]

What makes one puppy "unfair"?

The fact that we know that it is male.

No. You're conflating the general probability distribution with an observed fact about this one particular instance.

That's "gambler's fallacy."

We're not looking to figure out general probability distribution, we're trying to figure out something about this one particular instance.

Like you said, That's "gambler's fallacy."

No, the general probability distribution is EXACTLY what we're trying to figure out! Essentially, what they want to know is, if this scenario were to happen an infinite number of times, what percentage of the scenarios (that have at least one male) would have two males?

Hell, that right there is why the computer simulation works for a rough estimate. You test a bunch of times, throw away the tests that don't fit the scenario, and use the remaining results to calculate rough probabilities. It's not exact, but it's great for testing your math.

...

Sigh. I forgot how annoying internet arguments can be.

 

[Samirat]

No no no!

Where did the two puppies in the problem come from? How were they initially selected?

What do you mean "initially"?

He means "in the beginning." Like, how were the two puppies for the problem selected? Randomly, right? They represent a random pair, where at least one happens to be male. If you have a random pair, there is a certain likelihood that at least one will be male. This falls in that category. But it's information that doesn't change the fact that they are random puppies.