Also you might want to calculate the probability that LESS than half are chosen, since
P(0,1,2) are all zero. So you only need to calculate three values instead of 7.
Try thinking about the whole population of bottles.
X% are not defective.
Y% are defective.
Now you pick a bottle at random and drop it four times.
What can happen?
It could be defective and survive. (a%)
It could be not-defective and survive (b%)
It could be defective and break...
Here is a thin sketch of a proof. Then you can fill in the details and ask for help on steps you don't know how to do.
1) Take a Cauchy sequence c_n in l2. Construct a new sequence X by treating each coordinate as a Cauchy sequence in R (or C).
2) Show that X is in l2.
3) Show that c_n...
Well let's see. Someone has the gene 93% of the time, and when they do it's reported that they have it 99.9% of the time. So out of 100k people, 930 would have the gene but the test would say no.
7% of people don't have the gene, and it's reported that they do 10.3% of the time. So out of...
Categorize all the people into the following:
Has the gene/test says yes
Has the gene/test says no
No gene/test says yes
No gene/test says no
Figure out how many in each group. Then you should be home free to answer any questions about this test and population.
Well, you might want to look into Bayes' Theorem (try Wikipedia).
An intuitive way to do it is to create a large population, and then figure out what happens to it.
So suppose there are 100,000 people. How many of them actually have the disease? Suppose we test them all. How many are...
Hint: If you have a partition P_n, then each piece of a partition that contains a point where f is nonzero has an area of \frac{1}{n}. What you need to show is that for any \epsilon, there is an n such that no more than n\epsilon of the pieces contain a point where f is nonzero.
Try to do...
No, the question is asking for the chance that after 200 bombs were dropped, a particular block had not been hit.
You are right that P(not getting hit by a single bomb) = 99/100. So what's P(not getting hit by any of 200 bombs)?
I think the answer is 1007 turns. You have to force the opponent to an end to catch him.
WLOG suppose you know he is at a location x < 1006. Then you can force him to the nearest side by simply playing x+1. He must go to x-1 to avoid capture, so you next play x, and so on. Eventually he is...
Well, suppose you want to evaluate F(3). Then you need
\intop_1^9 \frac{10}{2+t^3} dt
But t is supposed to be between 0 and 4, right? So this expression is undefined.
Is this a homework question? If so, it should be posted in the homework section, and you should show what you have done so far, and people will give you hints or help.
http://scipp.ucsc.edu/~haber/archives/physics116A06/Sixways.pdf p.12-15 outlines the method.
The quick version:
Basically you draw a square with vertices \pm (N + 1/2) \pm i(N + 1/2) . Then you show that the integral of f(z)cot(\pi z) around the square goes to zero as N-> inf. So then you get...
The acceptable values for t are not the same as the domain of F (ie, the acceptable values for x). I can't tell from the last comment whether you were saying that you thought that or not.
This series can be summed using the method of complex residues.
Basically if you have a function f(z) that satisfies some weak criteria and an associated series as a function of n in the integers, you get:
\sum_{k=-\infty}^{\infty} f(n) = -\pi \sum res \left[\cot (\pi z) f(z) \right]...
Well, yes, it wouldn't be correct for you to set dr/dt (the rate at which the radius is increasing per time) equal to 1900 (the size of the patch at some point in time).
You want the rate at which the radius is increasing when the area is 1900. You have an expression for the rate. So evaluate...
Taking the derivative just guarantees that you can easily find a minimum or maximum of the function. It helps most when it's not that obvious where a function is below or above zero.
It doesn't matter that it's a parabola, and you can do it equally by guessing, plotting the function, or other...
Hi.
The formula for the residue of a function at z_0, a pole of order n is:
res(f;z_0) = \lim_{z \rightarrow z_0} \frac{1}{(n-1)!} \frac{d^{n-1}}{dz^{n-1}} (z-z_0)^n f(z)
Here the factorial term is just 1! = 1. So what we need is the derivative of (z-z_0)^2 f(z) .
So we get...
Let a = mk and b=nk, where n and m are integers, and k is some prime factor of a and b.
Claim: (a-b) is divisible by k.
(a-b) = (mk-nk) = k(m-n)
m-n is an integer because m and n are integers, hence (a-b) is divisible by k.
The reason we don't allow fractions here is just because when we are talking about prime numbers, we are only concerned with integers, because that's what makes a number "prime." I mean, if we allowed fractions, then every number would be composite, and in an infinite number of ways. That just...
a and b don't have opposite signs (necessarily), the function evaluated at those points has opposite signs.
Now you can choose any points you want, as long as they satisfy that condition. In this case, you have a parabola. So one way to find points that will be suitable is to take the...