No. If you consider the integers, there is no other number between 1 and 2. That does not mean that they are the same number.
Any proof that .999… = 1 has to involve limits in some way, because .999… is a limit.
Why should a proof based on the reals also work for the integers? The “there is no number between A and B” proof works because the reals are a continuum; the integers are not a continuum.
Though now that I think about it, being a continuum seems to be overkill… the same proof would work for the rationals as well (that is, A and B are the same if there is no number between them).
Anyway, the proof does implicitly contain a limit. Our sums go as 0.9, 0.99, 0.999, etc., and we observe that there is no number smaller than 1.0 that the sum will not eventually exceed as the index goes to infinity.
Well, perhaps realizing that there is no dirt filling the volume of the hole, he takes the question to ask for the surface area of the dirt in the hole?
A hunter walks 5 miles south of his cabin, then turns and walks 5 miles east where he shoots a bear. He then walks 5 miles north back to his cabin.
What is the color of the bear?
Clearly when speaking about .999…, we are not talking about integers.
That’s not what the question asked. However, a poorly formed question can produce a variety of potential answers. So an answer could be “X sq. ft., if you were asking about the surface area of the dirt surrounding the hole”
Or going back to a thread in GQ, restate the question in a reasonable form before answering.
Plaid. It was a scotch bear.
Now? Red and white.
There are no bears or cabins at the North Pole.
a hunter walked 5 miles south of his cabin, turned 360° and walked for another 5 miles. where did he end up?
It says to read everything before doing anything and the last step says to only write your name. The explicit instructions override the unspoken assumption.
It is conceivable that a polar bear could wander that far. Also, a cabin can be built just about anywhere.
I guess I don’t get this one. Aren’t the odds about 50% that the other is a girl?
2 kids means 4 possibilities: boy/boy, boy/girl, girl/boy, girl/girl. We know at least one is a girl so we can eliminate the first possibility. Of the three remaining possibilities, 2 are girls and one is a boy so the odds are 67% that the other is a girl.
You’re thinking, “There are two possibilities: either they have two girls, or a boy and a girl. So if one child is a girl, there’s a 50/50 chance that the other is a girl.”
The problem is, that’s like thinking that you a 50/50 chance of winning the lottery, because there are two possibilities, win or lose. In real life, one of those possibilities is more likely than the other, and the same is true with this question.
Consider this: When a family has a child, they have (more or less) a 50% chance of having a girl. So let’s imagine four families. We’ll give two of them boys to start, and two of them girls.
Family A: Kid 1 = Boy
Family B: Kid 1 = Boy
Family C: Kid 1 = Girl
Family D: Kid 1 = Girl
Now, each family decides to have a second child. Again, there’s about a 50% chance that each of these children will be a girl.
Family A: Kid 1 = Boy, Kid 2 = Boy
Family B: Kid 1 = Boy, Kid 2 = Girl
Family C: Kid 1 = Girl, Kid 2 = Boy
Family D: Kid 1 = Girl, Kid 2 = Girl
So now, look again at our question:
“Your new neighbor has two children.”
Fine, that could be any of these four families.
“At least one of them is a girl.”
Okay, that removes Family A.
“What are the odds the other one is a girl?”
Well, we have Family B with one girl, and the other is a boy. We have Family C with one girl, and the other is a boy. And we have Family D with one girl, and the other is a girl. So of three possible situations with one girl, only one has another girl. Therefore, the odds that the other child is a girl are 1/3.
ETA: Inner Stickler, I think you have it backwards - don’t you mean “2 are boys and one is a girl so the odds are 67% that the other is a boy.”?
It still doesn’t click with me. I’m looking at it like coin tosses. No matter the outcome or how many times I flip the coin, each toss has a 50% chance of landing “heads.”
it’s “at least one of them is a girl” and not “the first one is a girl”.
Sure, the next coin flip has a 50/50 shot at being heads. And the family’s next kid has a 50/50 shot of boy or girl, just like the first two kids did. But the information provided was that at least one is a girl. Had the person said “the first is a girl,” “the second is a girl,” or “only one is a girl,” then the probability of a brother would be 1/2.
But as it is, the girl can be first, second, or both. That’s where the 2/3 comes from. Saying “at least one” is subtly different than “one”.
Right. But just like with coin tosses, there are two different kinds of questions you can ask:
“What are the chances that any given toss will be heads?”
vs.
“What are the chances that all tosses will be heads?”
It’s like the typical trick question about coin tosses. It sets you up to *think *you’re considering the odds of the entire run of tosses, but then *asks *only about a single toss:
“I’ve just thrown 20 heads in a row. What are the chances that my next toss will be heads?”
The chances for the next toss are 50/50. It doesn’t matter what happened before. But if you change the question, you get a different answer:
“What are the chances of throwing heads 21 times in a row?”
That’s .5(.5)(.5)(.5)(.5)(.5)(.5)(.5)(.5)(.5)(.5)(.5)(.5)(.5)(.5)(.5)(.5)(.5)(.5)(.5)(.5), or 0.000000476837158203125 (if I did that correctly).
So the girl-children question works in the other direction. You *think *it’s asking, “What are the chances that any one child will be a girl?” but it’s really asking, “If you have two children, and don’t have two boys, what are the chances that they are both girls?” There is a 1/4 chance that, if you have two children, you will have two girls. You also have a 1/4 chance of having each of the following: a girl followed by a boy, a boy followed by a girl, and a boy followed by a boy. And since we’re disregarding the two-boy families for this question, there are three other possibilities. That leaves you with a 1/3 chance of having two girls.
Okay, I get it. That’s where I was confusing myself.