I got twelve by following it round, almost as if it were a clock, but then I noticed that there are other ways (e.g. left, up, left). I'm sure there's a way to be sure (this isn't really my area of expertise) but just a cursory attempt at a brute force solution seems to suggest to me that 16's a pretty good answer.
I'm not sure that's necessarily true. NN could be a woman and, therefore, the mother of "the son of NN". The father of "the son of NN" would then be her husband/spouse/whatever. Then the son of the father of NN would be NN's brother and, therefore, the uncle of "the son of NN" and the brother-in-law of "the father of the son of NN".
Presumably the use of the word "the" excludes the other possibility that "the father of NN" had many sons, and - therefore - one of them was NN's brothers.
Hmmmm....on second thought, maybe? I'm not sure. My original thinking was that when the first ± is +, so is the second one and when the first ± is -, so is the second one. So, for example:
When the first ± is +, so is the second one and when the first ± is -, so is the second one. So we have:
Hence the need for a ∓ sign. In this case we would have +(+5) or -(-5) and only these options. But it seems reasonable to be able to say:
So, yes, I think you're right. Ignore my second post. The first one still stands, though, I think
But yes, anonimnystefy is right. Sorry for confusing the matter further, the important thing to note is √x is always the positive, principal root, hence the need for the ± sign before the radical sign.
I would imagine it dates back to early geometry. If, say, you're trying to calculate the length of a hypotenuse, you're not interested in the negative values. I imagine this general precedence of the principal square root was incorporated into the notation when it was defined. Because, of course, we define our notation to be useful to us and easy to work with. But, without realising it, you've always been using the convention whenever you've gone:
If it weren't for the fact that the √ sign only referred to the principal value, you wouldn't need the ±, that would be implied by the √. Then you could just write:
It's just the way we learn to think about it conceptually
I agree, it was sloppy phrasing on my part as well. Although, bobbym, I think your post #35 was correct. 9 does have two square roots (the principal root being 3, the other being -3) the problem is that the notation √9 gives us the principal square root. This is why we have to write the ± sign before the radical.
Thus ± √9 = ±3 and √9 = 3.
But "the square root of nine is plus, or minus, three" is correct (or, perhaps I should say "the square roots of 9 are..."). It is a problem of formal notation that we have, I believe - if I have understood everything correctly.
Actually,only, but both 3 and -3 satisfy the equation .
Excellent technical observation, we should, really, say that:
However, it is true to say that the square root of 9 is plus or minus 3. The problem we have is that the sign √ refers to the principal square root only.
This does make the thing a little harder to understand, though
Suffice it to say that when we square root both sides of an equation, we must include the ± sign, as an equation of the form:
Has two solutions.
Yep, because a negative number times a negative number is a positive number, so:
Therefore, when we 'undo' the squaring, by square rooting, the answer could be positive, or negative. The square root of 4 is either 2, or -2. We can't tell.
Edit: this is why bobbym's original solution had two answers
Which is the same as:
Sorry, to quote:
"The probability that the kth integer randomly chosen from [1, d] will repeat at least one previous choice equals q(k − 1; d) above. The expected total number of times a selection will repeat a previous selection as n such integers are chosen equals:"
I can't see what to do with this, either, but I suppose you could incorporate the expected total number of collisions into your calculation, along with the probability of two collisions already calculated. I just don't know how you would do that. If you know of a computer model, though, I'd be very happy just to have a numerical answer, I can understand that it would be very difficult to calculate this algebraically
I think his suggestion was about 3 & 4, yes.
I was having a lot of problems with 1 & 2, but then someone pointed me in the direction of the formula:
Where n is the number of people in the group, k is the range of days (so k = 7 in the case where I'm trying to calculate the number of people required in the group for me to be certain that two of them have birthdays within a week of each other) and m = 365 (the number of days, excluding the 29th of February.)
Using this, I was able to get the same answers as you
I'm completely stuck on 3 & 4, though. I can understand that there isn't an algebraic solution to this problem, but is there a numerical one? The only advice i've been given is to try and make use of that formula, but I wouldn't know what to do with it!
Thank you bobbym, that's great. I've been playing with the numbers myself and I'm completely with you on 1 & 2 . 3 & 4 are utterly beyond me, though. Someone suggested the formula in the wikipedia article under collision counting as a possible approach to me, but i've no idea, really, what to do with it
Hi, I've been looking at the birthday problem (which is a statistical problem which aims at finding out the how many people you would need in a random group to be certain that two of them shared a birthday. Obviously the vacuous answer is 367, but as it turns out, there is a probability of 99% that two people will share a birthday in a group of just 57 and 50% in a group of just 23 (see: http://en.wikipedia.org/wiki/Birthday_problem)).
Okay, this is fair enough and very interesting, but I was trying to take the principle further. i wanted to find out how many people you'd need to be confident that two of them shared a birthday in the same week and then how many shared a birthday in the same month, then how many shared a birthday in the same two month period. Unfortunately, I got completely confused, so I came over here. I know this is 3 questions in one, but i imagine it's just number shunting.
I was also playing with the numbers for a friend and trying to work out the probability that in a random group of five people, two of them would share a birthday in the same month and, i suppose the final question i really have is: imagine a person has met 500 people in their life whom they've really had the chance to get to know. what is the probability that seven of those 500 share a birthday in the same two month period?
I know the last question is quite difficult, but those numbers aren't entirely plucked out of thin air, I was discussing with my friend the statistical significance of her friends sharing birthdays in broad ranges like this.
I've been thinking about this and I've realised I've not worded this at all clearly. Let me try and make this question intelligible:
Question 1: What is the smallest group of randomly selected people required such that the probability that two of them share a birthday within one week of each other is at least 75%?
Question 2: What is the smallest group of randomly selected people required such that the probability that two of them share a birthday within thirty days of each other is at least 75%?
Question 3: What is the smallest group of randomly selected people required such that the probability that seven of them share a birthday within sixty days of each other is at least 75%?
Question 4: In a group of 30 randomly selected people, what is the probability that seven of them will share a birthday within fifty days of each other?
Hmmm...that's what i was trying to do, well, I really need to get off to bed, but for what it's worth, our old friend Wolfram gives, as its answer to
Which I can get from my answer, so I guess it's just a question of laws of logs, I don't suppose anyone has any ideas?