TimD1
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Why is there a unique solution to the frog puzzle?
6 votes

I have been unable to come up with anything resembling a formal proof, but I have noticed several things that may help someone do so. At each step, you must choose between moving a frog of color $A$ ...

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Understanding Bernoulli Trials
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4 votes

$p^k$ represents the probability of exactly $k$ consecutive successful trials $q^{n-k}$ represents the probability of exactly $n-k$ consecutive failed trials However, we don't care about the order. ...

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$12$ people $p_1, .. , p_{12}$ divided into $3$ groups, what is the probability that $p_i$ and $p_j$ are in the same group?
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3 votes

In this problem it's easier to first view it as three possible situations $\frac{3}{12}\times\frac{2}{11}$ Chance that they are both placed in first group $\frac{4}{12}\times\frac{3}{11}$ Chance ...

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Converting an equation from Cartesian to Polar form?
2 votes

The easiest way to remember the formulas for converting polar to rectangular coordinates and vice versa is to draw the right triangle at the origin with sides $x$ and $y$, hypotenuse $r$, and angle $\...

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Period of a Wind Turbine Blade
2 votes

In short, you are correct. Here's a quick explanation of what each part of the equation physically represents, and how it would affect the graph of a cosine equation. $$H(t)= A + B \cos\left(Ct\right)....

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Is the Below Theorem true:
2 votes

Yes, I think you're correct. Alternatively, once you have established that there are $2n-2$ possible lines, you could argue that each line can either be drawn or not ($2$ choices), and each set of ...

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Maximize the probability of success to pass the exam
2 votes

First, note that in order to correctly solve "two consecutive tasks", you must solve the middle task, and then either the first or last (or both) as well. In other words, you must pass the middle task ...

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How do I eliminate left recursion in discrete math
1 votes

As you mention, the formula for removing left-recursion (and replacing it with right-recursion) involves replacing all instances of this rule: \begin{gather} A \rightarrow A\alpha\ |\ \beta\\ \end{...

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How to write a grammar that every 0 is followed by at lease one 1 in Discrete math
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1 votes

Here is a context-free grammar for the language you describe, assuming the alphabet is $\{0,1\}$: \begin{gather} S \rightarrow ST1\ |\ \epsilon\\ T \rightarrow 0\ |\ \epsilon\\ \end{gather} This ...

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The probability that two balls with the label-sum $4$ are picked
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1 votes

Your reasoning in the question is correct, however your continued analysis in the comment is not. In this case, order is unimportant because the questions asks for the probability that the sum of the ...

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Alternative perspective on mean value theorem in one-dimensional space
1 votes

What the mean value theorem basically states is that on a continuous function, given two points $(x_1, y_1)$ and $(x_2, y_2)$ where $x_2 > x_1$, the function's derivative somewhere on the interval ...

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Two questions about beginning calculus.
0 votes

To solve the first question, use the equation for a line: \begin{gather*} y - y_1 = m(x - x_1) \end{gather*} For the second question, In order for $f(x)$ to be continuous at $x = 8$, \begin{gather}...

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Sat math section simple mistakes
0 votes

Estimate. Before solving a problem, quickly approximate what you would expect the ultimate solution to be. Before answering, ask yourself, it this value around what I initially expected? Does it make ...

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How do I simplify this regex?
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0 votes

To construct a DFA which accepts all strings not containing 110, you can't just use the negation of a DFA which does accept 110. You must use the negation of the DFA which accepts exactly those ...

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Probability with union bound
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0 votes

Hint: Consider all possible placements of equilateral triangles (with side length 1000) placed on the globe. If none of these triangles have all three vertices in the water, then at least one out of ...

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Subject GRE 0568 exam Q.55
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0 votes

The prime factors of $10$ are $2$ and $5$, so $10$ can be rewritten as $2^15^1$. Since the prime factorization of any number is written as $2^a3^b5^c7^d11^e\dots$, the number of trailing zeros it has ...

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The logic behind this explanation
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0 votes

In normal division, $\frac{34}{5} = 6$ remainder $4$. Or, you could re-express it by dividing the remainder by the original divisor as $6+\frac{4}{5}$. This corresponds directly to what is shown in ...

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Binomial Random Variable Problem?
0 votes

You are correct. The probability that Player A wins $3$ matches and loses $2$ matches is $p^3q^2$, and there are ${5 \choose 3}$ ways in which this could happen. Therefore, the probability that Player ...

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What is the probability of escape (read the full question to understand, its probability)
0 votes

The probability of escape is $\frac{2}{3}$. There are six successful ways to escape: He chooses door 1 He chooses door 2, then 1 He chooses door 3, then 1 He chooses door 2, then 2, then 1 He ...

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How to evaluate length of side of triangle
0 votes

This question expects you to already know the properties of two types of triangles. First, let the intersection of AC and the altitude of triangle ABC be called point D. In a 30-60-90 triangle, the ...

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