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2d
comment Looking to understand proposition related to the fundamental theorem of algebra
This is the FTA, and the proof is not easy. Another version of the FTA is that any non-constant polynomial with complex coefficients has a root in the complex numbers. From that lemma you can prove the FTA as stated above quite simply. But proving the lemma is difficult.
2d
comment Looking to understand proposition related to the fundamental theorem of algebra
You have called it by its usual name in the title. Another quick way of stating it is that the complex numbers are an algebraically closed field.
2d
comment Probability of the simultaneous failure of two components where one component would take the full load if only one failed instead.
There is something puzzling about the first solution. We have non-failure if neither beam fails, or one fails and the other holds. The probability exactly one fails is $2P_{1/2}(1-P_{1/2})$, and not $2P_{1/2}$. If $P_{1/2}$ is small, the $2P_{1/2}$ gives a reasonable first-order approximation.
2d
answered How do I evaluate $\lim_{n\rightarrow \infty}\frac{1}{\sqrt{n}}\sum_{k=1}^{n}\frac{1}{\sqrt{k}}$?
2d
comment How many ways shuffle $n_1$ and $n_2$ balls when we but them together?
Imagine $n_1+n_2$ slots where the balls will go. We need to choose the $n_1$ slots where the white balls will go. This can be done in $\binom{n_1+n_2}{n_1}$ ways.
2d
reviewed Leave Open Find the smallest possible value for: $a+b$
2d
reviewed Leave Open Help with the conditional probability in this problem
2d
comment Solving an equation that contains a logarithm
This is decreasing from $0$ to $1$, then increasing, In general there will be two $x$ for every $y$. And, as has already been observed, we cannot solve using elementary functions.
2d
comment Probability of drawing >18 when drawing 3 cards
Doable, but exceedingly messy. I would suggest doing a large simulation, and use the resulting frequencies as a table of values.
2d
revised Erin rolls 4 four-sided dice all at once, then can roll a subset of her choosing a 2nd time. What is the probability of getting all the same number?
added 440 characters in body
2d
answered Erin rolls 4 four-sided dice all at once, then can roll a subset of her choosing a 2nd time. What is the probability of getting all the same number?
2d
comment Erin rolls 4 four-sided dice all at once, then can roll a subset of her choosing a 2nd time. What is the probability of getting all the same number?
The only thing that makes sense is that she leaves certain carefully selected dice as they fell, and tosses the rest, hoping that at the end all $4$ up faces will be the same,
2d
comment ***M>N*** Find the ratio of M to N and hence find two possilbe sets of lengths for the sides
Seems kind of hard to find two,
2d
comment Volumes of revolution (annular cross sections)
What you asked for has just been done in the answer by reluctant mathematician.
2d
comment Volumes of revolution (annular cross sections)
Use the method of Cylindrical Shells. It can be done by slicing, but is more complicated that way.
Jul
4
comment Rotation of $y=x(1-x)$ about the $x$-axis
You are welcome.
Jul
4
comment Probability dealing with Odds
Let $p$ be the probability. Then $\frac{p}{1-p}=\frac{a}{b}$. Solve for $p$.
Jul
4
comment Rotation of $y=x(1-x)$ about the $x$-axis
This is the first volume calculation you met, $\int_0^1 \pi x^2(1-x)^2\,dx$. Expand and integrate.
Jul
4
revised Dice Roll Probabilities
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Jul
4
comment Probability of picking all white marbles?
There are $\binom{n}{k}$ equally likely ways to pick $k$ marbles. There are $\binom{5}{5}\binom{n-5}{k-5}$ ways to pick $5$ white and the rest non-white.