= 27d567bc2d48d9f40e9ccc20ea370b15)
You're not. Remember, you're not taking 80 of the 10,000 women in the population. You're only taking 80 of the 100 women with breast cancer. Likewise, it's not 9.6% of all the women, it's 9.6% of the women who don't have breast cancer, or 950/9900. The wording of the problem already took the base rates into count, so when you're plugging the real numbers in, you are automatically taking the base rates into account. By giving you 80/100 and 950/9900, Eliezer already did the division for you.
....Oh.
Well, thanks Owen, Swimmy. I now understand Bayes Theorem significantly more than I did a half hour ago. :)
In response to
Trouble with Bayes Theorem? (The actual math is confusing)
You're forgetting the "base rate" in your calculation: the actual rate of cancer in the population. What you should really be taking the ratio of is (the fraction of all women that have cancer and test positive) / (the fraction of all women that test positive, whether or not they have cancer). In percentages, that's
(80% of the 1% of women who have cancer, who correctly test positive) = 0.8 * 0.01.
divided by
(80% of the 1% of women who have cancer, who correctly test positive) together with (9.6% of the 99% of women who don't have cancer, who test positive anyway) = 0.8 * 0.01 + 0.096 * 0.99.
So the ratio is (0.8 * 0.01) / (0.8 * 0.01 + 0.096 * 0.99), and that does equal 0.078.
Thanks. I'm pretty sure I understand now. Although I'm not sure why I get the correct answer when I'm working with the actual numbers and not percentages when I do the math wrong.
But when I do the math like you wrote, I get the right answer for the precentages. So I get that part. But aren't I ignoring the base rate in the actual numbers one? Or no?
Trouble with Bayes Theorem? (The actual math is confusing)
This is probably going to sound utterly ridiculous, but I have a sad confession.
I've read Yudkowsky's post on Bayes' Theorem (http://yudkowsky.net/rational/bayes) five times. I've written down the equation. Tried to formulate an answer.
I still don't understand it. That being said, I've lived my entire life under the false mentality that maths is boring and painful, and it's just recently I've tried to actually understand the concepts I learn in school, and not just temporarily memorize them for the next exam.
Here's the problem, on Yudkowsky's post:
"1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?"
When Eliezer changes the percentages to real numbers:
"100 out of 10,000 women at age forty who participate in routine screening have breast cancer. 80 of every 100 women with breast cancer will get a positive mammography. 950 out of 9,900 women without breast cancer will also get a positive mammography. If 10,000 women in this age group undergo a routine screening, about what fraction of women with positive mammographies will actually have breast cancer?"
When I see this equation, I can properly make the answer come out to 7.8 percent. I do this, by taking the 80 women, and dividing 80 women by the 80 women plus the 950 women, so 80/80+950 (or 80/1030=.078). So I get 7.8%, which should be the right answer.
But when I try to do the same with percentages, it all gets sort of screwy. I take the 80 percent of women (.8) divided by that same 80 percent (.8) plus 9.5 percent of women without cancer who test postive for it (.095). So I get .8/.8+.095=89%.
I feel like I'm making a really, really stupid error. But I just don't know what it is. >_>
In response to
Subjective Realities
I think The Simple Truth, from the Map and Territory sequence has the answer to your question. It addresses this exact argument, in fact.
Whoever thumbed up my comment about not understanding.... Why?
XD If someone doesn't understand something, I'm not going to slap them on the back and tell them "Good job."
Thank you (and others who have posted) for helping me with this silly argument.
:) I'll let you know how things turn out. He's quite clever at thinking on the spot.
Update:
I confronted him with many of these arguments.
He still expects that people will exist in the future.
I think I've figured it out. This is what is wrong with his belief.
- It doesn't pay rent.
Even if he's right, and people stop existing when we stop percieving them, it still won't change how we behave, or what we're expecting to happen. He expects to see his friends later, he just says he can't prove they exist at the moment. (I asked him about memories, he said that we still have memories of dead people, are they alive?)
- It's not falsifiable. It doesn't constrain experience, I can't use this new idea as a model for future information. It's utterly useless because it permits anything to happen.
When I think about it, it's like the tree dropping in the forest analogy. We're not anticipating different experiences. We both expect to see our mothers later. And we can both agree that we cannot percieve people when they aren't in front of us.
He's just choosing a rather complicated way to state the obvious. At least this is what I'm getting.
Am I right, guys?
In response to
Subjective Realities
I mean, he challenged me to prove that my mother existed, without seeing her. Obviously I couldn't.
You didn't have a phone number you could reach her with? You didn't have any memories of her? These are all sufficient evidence for her existence. And sufficient evidence is what we call "proof".
At the very least, you could tell him "I bet you a thousand dollars that I will later see my mother, when I go home" If he doesn't accept the bet, that's evidence he himself believes in the existence of your mother, as he expects to lose your bet.
His words babbling about "existence" mean nothing -- as his arguments connects to neither the present and the evidence you have now (your memories of your mother), nor the anticipated future. It's just babbling nonsense.
Thank you (and others who have posted) for helping me with this silly argument.
:) I'll let you know how things turn out. He's quite clever at thinking on the spot.
In response to
Subjective Realities
I think The Simple Truth, from the Map and Territory sequence has the answer to your question. It addresses this exact argument, in fact.
I've already read that, and I still don't understand.
Subjective Realities
So I have a friend who I sit next to in class, and we talk about philosophy. Well today, he brought up that when people leave your presense, and you can't observe them any longer, you no longer have proof they exist.
Well I pointed out that it would violate the conservation of mass law, right?
So then, with a bit more prodding, I figured out that by "no longer exist", he means they exist in their world, but they no longer exist in mine. So basically you can't prove that anyone exists unless they're directly in front of you.
I'm really not certain how to go about answering this question. I mean, he challenged me to prove that my mother existed, without seeing her. Obviously I couldn't.
Is he right? Or is there some flaw in his argument, some fallacy that I'm missing?
I went through a few of the Sequences, and the closest article I could find was about not believing in the invisible. But in this case, he doesn't literally (I think) believe they just vanish, he believes they enter alternate universes that are selected when I come in contact them again.
My mind is boggled. I also apologize if this is dumb question, and it's common knowledge or has already been answered, and to my credit, I did make an attempt to figure out the answer before bothering you all. Thanks.
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I know it now makes more sense to you now, but I want to point out that reality isn't school, and nobody is going to take marks off for using actual numbers or ratios instead of percentages (the 'pure' way that the teacher prefers or what-have-you).
A calculator more reliably gets me the answer than mental arithmetic, and so I use a calculator at work even though it seems lazier than doing it in my head - in the same way, if ratios and actual numbers more reliably let you use Bayes Theorem than percentages, use actual numbers and all the people who think it's purer to use percentages be damned.
I'm awfully glad to here that, I'm not a big fan of percentages... Real numbers just come easier to me, I suppose.
Once I figure out the formulat itself, then I feel comfortable using a calculator, but I hate using a calculator if I don't understand the mental math to begin with.