I'm studying probability and am currently stuck on this question:
Let's say we have n distinct dice, each of which is fair and 6-sided. If all of these dice are rolled, what is the probability that there is at least one pair that sums up to 7?
I interpreted the above as being equivalent to the...
Homework Statement
Here is an exercise I came across while studying MaxFlow / MinCut and I'm rather stumped:
Designate G = (V, E) as an undirected graph that has at least two vertices. Every edge e on this graph has capacity ce. Pick two arbitrary vertices from G and label them s and t...
Homework Statement
You are given some undirected graph G = (V, E), along with a set S which consists of 0 or more pairs of G's edges.
As an example, a complete graph on 3 vertices (a triangle, basically) would be described as follows: G = (V, E) = ({v1, v2, v3}, {v1v2, v2v3, v1v3}). A set S...
Homework Statement
1. Give a language L that cannot be decided by a TM using space O(log n) and time less than n on inputs of length n. The language L should be decidable by some TM. Assume the TM has a binary input alphabet.
Homework Equations
Undecidability, Turing Machines, Languages...
Homework Statement
We'll define a binary tree as a tree where the degree of every internal node is exactly 3. Show that the set of all binary trees is countable.
Homework Equations
A set is countable if it is finite or there is a one-to-one correspondence with the natural numbers.
The Attempt...
Question: Suppose you have a finite state machine that accepts only strings with an equal number of zeros and ones. Show that you can then construct a finite state machine that accepts only strings of the form 0^n 1^n, i.e., an arbitrary finite number of zeros followed by the same number of...
Homework Statement
Alice proposes to Bob the following game. Bob pays one dollar to play. Fifty balls marked 1, 2, . . . , 50 are placed in a big jar, stirred around, and then drawn out one by one by Zori, who is wearing a blindfold. The result is a random permutation (let's call it s) of the...
Homework Statement
An ice cream shop has a special on banana splits, and Xing is taking advantage of it. He’s astounded at all the options he has in constructing his banana split:
· He must choose three different flavors of ice cream to place in the asymmetric bowl the banana split is served...
OK, now I'm really confused then. Because sometimes using the surface integrals get me the correct answer but using Divergence Theorem doesn't, and vice versa. Below is a VERY similar question that I solved correctly using surface integrals...and I can't for the life of me see what is...
Homework Statement
The problem is given in the attached file.
Homework Equations
Divergence theorem, flux / surface integral
The Attempt at a Solution
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As you can see I got the question correct using Divergence theorem. But I wanted to make sure that I could arrive at the same answer...
Wait, so...I gather you're saying that ##\sum(-1)^{n+1}*(1-n^{1/n})=\sum(-1)^n*(n^{1/n}-1)##, so when we consider the positive term ##(n^{1/n}-1)##, it can be re-written as:
##\frac{(n^{1/n}-1)*n}{n}##, aka:
##\frac{\frac{(n^{1/n}-1)}{\frac{1}{n}}}{n}##?
And from here we do a...
Homework Statement
Does the following series converge or diverge? If it converges, does it converge absolutely or conditionally?
\sum^{\infty}_{1}(-1)^{n+1}*(1-n^{1/n})
Homework Equations
Alternating series test
The Attempt at a Solution
I started out by taking the limit of ##a_n...
Two things wrong:
1) As another poster noted, the PE is incorrect. Note that the problem says the crate is 350 Newtons, not 350 kg.
2) Wnc is not just -50N. There's something missing there. Remember the general equation for Work and check your units.
The difference between net Force and net Torque is something you should internalize because doing so also helps you understand things like why we have separate equations for angular momentum vs. linear momentum, or translational KE vs. rotational KE.