inclusionAI/Ring-lite-rl-data huggingface dataset:
6. Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7 . Play continues until either player wins, or else indefinitely. If Nathaniel goes first, determine the probability that he ends up winning.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{11}}
Find all pairs $(x,y)$ of real numbers such that $ |x|+ |y|=1340$ and $x^{3}+y^{3}+2010xy= 670^{3}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x,y)={(1005,-335),(-335,1005),(-670,-670)}}
Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{223}{286}}
Consider a Markov process on the state space $S = \{1,2\}$ with a positive transition matrix, meaning all entries are strictly positive. Let $S_\infty = \times_{t=1}^\infty S$ be the underlying probability space, and let $\mathbb{P}$ be the probability measure on $S_\infty$ induced by the Markov transition matrix. For each $n$, define $F_n = \{ s \in S_\infty : \text{there are at most $n$ consecutive occurrences of $1$ or $2$} \}$. Determine if $\lim_{n \to \infty} \mathbb{P}(F_n) = 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a matrix $A=(a_{ij})_{n\times n}$, let $C_{i,j}$ be the cofactor in position $(i,j)$. By the determinant formula, we have \(\det A=\sum_{i=1}^n a_{i,1}C_{i,1}.\) Evaluate the expression \(\sum_{i=1}^n a_{i,1}C_{i,2}.\) Must this expression evaluate to zero?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a square matrix $A = (a_{ij})_{n \times n}$, evaluate whether the expression $\sum_{i=1}^n a_{i,1}C_{i,2}$ equals zero, where $C_{i,2}$ are the cofactors of the elements in the second column.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
For each integer $n\geq 2$, let $S_n$ be the sum of all products $jk$, where $j$ and $k$ are integers and $1\leq j<k\leq n$. What is the sum of the 10 least values of $n$ such that $S_n$ is divisible by $3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$197$}
Determine the boundary of the set \(\mathbb{Q}^n\) when considered as a subset of \(\mathbb{R}^n\) with the standard topology. Is it true that \(\partial \mathbb{Q}^n = \emptyset\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\partial \mathbb{Q}^n = \mathbb{R}^n}
Subsets $S$ of the first 3$5$ positive integers $\{1, 2, 3, ..., 35\}$ are called [i]contrived [/i] if $S$ has size $4$ and the sum of the squares of the elements of $S$ is divisible by $7$. Find the number of contrived sets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8605}
Two congruent equilateral triangles \(\triangle ABC\) and \(\triangle DEF\) lie on the same side of line \(BC\) such that \(B\), \(C\), \(E\), and \(F\) are collinear. A line intersects \(\overline{AB}\), \(\overline{AC}\), \(\overline{DE}\), and \(\overline{EF}\) at points \(W\), \(X\), \(Y\), and \(Z\), respectively, with the conditions \(\frac{AW}{BW} = \frac{2}{9}\), \(\frac{AX}{CX} = \frac{5}{6}\), and \(\frac{DY}{EY} = \frac{9}{2}\). Determine the ratio \(\frac{EZ}{FZ}\) in the form \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers, and find \(m + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
The surface area of a sphere inscribed in a regular truncated triangular pyramid is related to the total surface area of the pyramid as \(\pi : 6 \sqrt{3}\). Find the angle between the lateral face and the base plane of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arctan 2}
In how many ways can we change the sign $\ast$ with $+$ or $-$, such that the following equation is true?

\[1 \ast 2 \ast 3 \ast 4 \ast 5 \ast 6 \ast 7 \ast 8 \ast 9 \ast 10 = 29\]


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine whether the limit \( \lim_{x\to-\infty}\sum_{n=1}^\infty\lambda(n)\frac{x^n}{\Gamma(n)} \) is finite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the behavior of the function \( f(a) = (\sqrt{a})^{f(\sqrt{a})} \) at \( a = 2 \). Does it converge to a specific value, and if so, what is that value?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the average of all positive integers that have three digits when written in base $5$, but two digits when written in base $8$? Write your answer in base $10$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Determine whether the condition $f^{(a)}(0) = 0$ for all $0 \leq a \leq n$ implies that $\lim_{x \to 0} \frac{f(x)}{x^n} = 0$, where $f^{(n)}$ denotes the $n$-th derivative and $f^{(0)} = f$. Assume that all $f^{(a)}$ are defined in an interval around $0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $d(n)$ be the number of all positive divisors of a natural number $n \ge 2$.
Determine all natural numbers $n \ge 3$ such that $d(n -1) + d(n) + d(n + 1) \le  8$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3,4,6
}
Let  $\Gamma_1, \Gamma_2, \Gamma_3$  be three pairwise externally tangent circles with radii  $1,2,3,$  respectively. A circle passes through the centers of  $\Gamma_2$  and  $\Gamma_3$  and is externally tangent to  $\Gamma_1$  at a point  $P.$  Suppose  $A$  and  $B$  are the centers of  $\Gamma_2$  and  $\Gamma_3,$  respectively. What is the value of  $\frac{{PA}^2}{{PB}^2}?$ *Proposed by Kyle Lee*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{8}{15}$}
Determine whether the sequence \((y_n)\) converges in the space \(L^1(0,1)\), where \(y_n(t) = \frac{t^n}{n^2(1+t^2)}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Starting on a triangular face of a right triangular prism and allowing moves to only adjacent faces, how many ways can you pass through each of the other four faces and return to the first face in five moves?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine for which \( n \) positive integer the equation: \( a + b + c + d = n\sqrt{abcd} \) has positive integer solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n=1,2,3,4}
Determine whether the Weyl tensor is always zero or undefined in two dimensions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine whether the field extension \( \mathbb{Q}(\cos(\theta/2)) \) over \( \mathbb{Q}(\cos(\theta)) \) is finite for \( \theta \in \mathbb{R} \). If it is finite, find its degree.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine whether the infinite product \( \prod_{i=1}^{\infty}\left(\frac{p_{i}-1}{p_{i}}\right) \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
We define $N$ as the set of natural numbers $n<10^6$ with the following property:

There exists an integer exponent $k$ with $1\le k \le 43$, such that $2012|n^k-1$.

Find $|N|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1988}
Determine whether the sequence \( y_n = \frac{\sqrt[n]{\prod\limits ^{n}_{k=1}( 2n+k)}}{2n} \) converges, using the logarithmic version of the Cesàro-Stolz theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{8e}}
Determine if there exists a number \(a\) such that all derivatives of \(x^x\) are positive on the interval \((a, \infty)\). If such a number exists, find the smallest possible value of \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the values of \( k \) for which the matrix \( \begin{bmatrix}0&1&0\\0&0&1\\0&-k-7&k+8\end{bmatrix} \) is not diagonalizable over the complex numbers \( \Bbb C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7, -6}
Determine the rate of convergence of the sequence $e^{-n}$ as $n \to \infty$. Find a sequence $\beta_n$ that converges to 0 such that $|e^{-n} - 0| \leq K|\beta_n|$ for some positive constant $K$. Express the rate of convergence using $O(\cdot)$ notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(e^{-n})}
Let $X$ be a non-empty set. Consider the power set $\mathcal{P}(X)$. Define an edge between $a, b \in \mathcal{P}(X)$ if their symmetric difference is a singleton, i.e., $\textrm{card}((a\setminus b) \cup (b\setminus a)) = 1$. For a finite set $X$, the graph has chromatic number 2. Determine if the graph still has chromatic number 2 when $X$ is infinite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $f:\mathbb{R} \rightarrow\mathbb{R}$ be a continuous function at $x_0\in\mathbb{R}$. Additionally, $f'(x)$ is defined for every $x\in(x_0-\delta,x_0+\delta)$, where $x_0\notin(x_0-\delta,x_0+\delta)$. Given that $\lim_{x \to x_0} f'(x)=L$, determine if $f'(x)$ is continuous at every $x\in(x_0-\delta,x_0+\delta)$, where $x_0\notin(x_0-\delta,x_0+\delta)$. Answer with 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
Given that Maria scored $5, 2, 4, 3, 6, 2, 7, 4, 1,$ and $3$ goals in her first $10$ soccer matches, her total score after $11$ matches is divisible by $11$, and her total score after $12$ matches is divisible by $12$. Find the product of the number of goals she scored in the eleventh and twelfth matches.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Determine the maximum order of a permutation in the symmetric group $S_{10}$. Additionally, find the smallest positive integer $n$ such that a bijection $f: \{1, 2, 3, \ldots, 10\} \rightarrow \{1, 2, 3, \ldots, 10\}$ satisfies $f^n = f$ but $f^k \neq f$ for all $1 \leq k < n$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Determine whether the integral \( \int_{-1}^1 e^{-x^{4}}\cdot(1+\ln(x+\sqrt{x^2+1})+5x^3-4x^4)\,dx \) can be evaluated.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{e}}
Let  $x_1$ ,  $x_2$ , and  $x_3$  be the roots of the polynomial  $x^3+3x+1$ .  There are relatively prime positive integers  $m$  and  $n$  such that  $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$ .  Find  $m+n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The complete graph of \( y=g(x) \), which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is \( 1 \)).

What is the sum of the \( x \)-coordinates of all points where \( g(x) = x+2 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \( H \) be the space of all absolutely continuous functions \( f:[0,1] \rightarrow \mathbb{C} \) such that \( f(0)=0 \) and \( f' \in L^2(0,1) \), with the inner product defined by \( \langle f, g \rangle = \int_0^1 f'(t) \overline{g'(t)} \, dt \) for \( f, g \in H \). For a fixed \( 0 < t \le 1 \), define the linear functional \( L: H \rightarrow \mathbb{C} \) by \( L(h) = h(t) \). Determine the norm \( \|L\| \) of this functional and find the vector \( h_0 \in H \) such that \( L(h) = \langle h, h_0 \rangle \) for all \( h \in H \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{t}}
Find the set of real numbers \( \lambda \) for which the rank of the matrix \( \begin{bmatrix} -3 & 5 & -5 \\ -4 & 6 & -5 \\ -4 & 4 & -3 \end{bmatrix} - \lambda I_3 \) is less than 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{-3, 1, 2\}}
The distance between the $5^\text{th}$ and $26^\text{th}$ exits on an interstate highway is $118$ miles. If any two exits are at least $5$ miles apart, then what is the largest number of miles there can be between two consecutive exits that are between the $5^\text{th}$ and $26^\text{th}$ exits?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
18
}
The rectangle $ABCD^{}_{}$ below has dimensions $AB^{}_{} = 12 \sqrt{3}$ and $BC^{}_{} = 13 \sqrt{3}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P^{}_{}$. If triangle $ABP^{}_{}$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overline{DP}$, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid.
[asy] pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6); draw(B--C--P--D--C^^D--A); filldraw(A--P--B--cycle, gray, black); label("$A$", A, SE); label("$B$", B, NE); label("$C$", C, NW); label("$D$", D, SW); label("$P$", P, N); label("$13\sqrt{3}$", A--D, S); label("$12\sqrt{3}$", A--B, E);[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{594}
Let \(X_t\) be a Brownian motion and \(a < 0\). Define the stopping time \(T_x = \inf\{t \geq 0: X_t + at \geq x\}\). If \(Y = \sup_{t \geq 0}(X_t + at) = x\), determine whether \(T_x < \infty\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{T_x < \infty}
How many natural numbers $n$ less than $10^{6}$ are there such that the sum of the digits of $n$ is even, and the sum of the digits of $(n+1)$ is also even


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45454}
Let $H$ be a subgroup of an abelian group $A$. Suppose $a_0 \in A$ satisfies $2a_0 = h_1 + h_2$ for some $h_1, h_2 \in H$. Does this imply that $H - a_0 = a_0 - H$? Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
Let $(X_n)_n$ be a sequence of random variables in $L^2$. Suppose there exists a constant $M$ such that for all $n$, $$P(X^2_n \ge M) < P(X^2_n \ge E(X^2_n)).$$ Determine whether it follows that $E(X_n^2) \le M$ for all $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E(X_n^2) \le M}
Let \( p,q,r \) be 3 prime numbers such that \( 5 \leq  p < q < r \) .
 Knowing that \( 2{p}^{2} - {r}^{2} \geq  {49} \) and \( 2{q}^{2} - {r}^{2} \leq  {193} \) ,find
 \( p,q,r \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
p=17, q=19, r=23
}
Let  $P$  be a  $2019-$ gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of  $P$  a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ( Every two adjacent knots in a cycle must be on the same diagonal and on every diagonal there are at most two knots from a cycle.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2018}
Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = x^2/16$.  The vertices of the six parabolas are evenly spaced around a circle.  The parabolas open outward with their axes being extensions of six of the circle's radii.  Parabola $P_1$ is tangent to $P_2$, which is tangent to $P_3$, which is tangent to $P_4$, which is tangent to $P_5$, which is tangent to $P_6$, which is tangent to $P_1$.  What is the diameter of the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Let the radius of the incircle of an equilateral triangle \( ABC \) be 2 with center \( I \). If point \( P \) satisfies \( PI = 1 \), what is the maximum value of the ratio of the areas of \( \triangle APB \) to \( \triangle APC \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 + \sqrt{5}}{2}}
Derive the expression for \(xz_x + yz_y\) in terms of polar coordinates \((r, \phi)\), given that \(z(x,y)\) is a function of \(x\) and \(y\), where \(x = r\cos\phi\) and \(y = r\sin\phi\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r z_r}
Let us consider all rectangles with sides of length $a,b$ both of which are whole numbers. Do more of these rectangles have perimeter $2000$ or perimeter $2002$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
More rectangles have perimeter 2002.

}
Masha has three identical dice, each face of which has one of six different prime numbers with a total sum of 87.

Masha rolled all three dice twice. The first time, the sum of the numbers rolled was 10, and the second time, the sum of the numbers rolled was 62.

Exactly one of the six numbers never appeared. What number is it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
You plot weight $(y)$ against height $(x)$ for three of your friends and obtain the points $(x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3})$. If $x_{1} < x_{2} < x_{3}$ and $x_{3} - x_{2} = x_{2} - x_{1}$, which of the following is necessarily the slope of the line which best fits the data? "Best fits" means that the sum of the squares of the vertical distances from the data points to the line is smaller than for any other line.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\dfrac{y_{3}-y_{1}}{x_{3}-x_{1}}$
}
If there exists $x\in \left[1,+\infty \right)$ such that the inequality ${(1+\frac{1}{x})}^{x+a}≥e$ holds for $x$, then the range of real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[\frac{1}{\ln(2)} - 1, +\infty\right)}
The ratio of the number of students from grades 1 to 6 in Wuyang Primary School is 30:29:28:27:26:25. After the current sixth graders graduate, 372 new first graders are admitted, and the remaining students are promoted to the next grade. This results in the ratio of the number of students in the lower grades (grades 1, 2, 3) to the higher grades (grades 4, 5, 6) being 10:9 at the start of the new term. What is the total number of students in Wuyang Primary School at that time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2052}
On Halloween 31 children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.

"Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes.

"Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes.

"Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes.

How many pieces of candy in all did the principal give to the children who always tell the truth?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7
}
What is the greatest number of two-digit numbers that can be written in a sequence such that any two adjacent numbers are not coprime, while any two non-adjacent numbers are coprime?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A borrower takes out a loan of $2000 for two years. Construct a sinking fund schedule if the lender receives 10% effective annual interest on the loan, and the borrower makes semiannual deposits into a sinking fund earning 8% interest convertible quarterly. Calculate the amount of each deposit needed to replace the loan amount.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{470.70}
Three distinct integers are selected at random between $1$ and $2016$, inclusive. Which of the following is a correct statement about the probability $p$ that the product of the three integers is odd?



Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$p<\dfrac{1}{8}$
}
Determine whether the initial value problem \( \frac{dy}{dx} = 2y^{1/2} \) with \( y(0) = a \) admits infinitely many solutions for \( a = 0 \) or for \( a \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a = 0}
A line passing through the origin $O$ intersects the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ at points $A$ and $B$. The line passing through $A$ perpendicular to $AB$ intersects $C$ at point $D$. Line $DB$ intersects the $y$-axis at point $E$. If $\overrightarrow{OB} \cdot \overrightarrow{OE} = 2|\overrightarrow{OE}|^{2}$, then the eccentricity of $C$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{2}}
Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$, $PB=6$, and $PC=10$.  To the nearest integer the area of triangle $ABC$ is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79}
Find the smallest natural number n such that for all integers $m > n$ there are positive integers $x$ and $y$ for which the equality 1$7x + 23y = m$ holds
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
392.

}
Determine the determinant of the matrix expression $ACB + BCA$, where $A$ and $B$ are $n \times n$ diagonal matrices, and $C = cc^T$ is another $n \times n$ matrix. Can this determinant be expressed solely in terms of the determinants of $A$, $B$, and $C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
6. When four dice are rolled randomly, the probability that the product of the four numbers obtained is a perfect square is $\qquad$ .


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{25}{162}}
Determine whether the number of conjugacy classes of the group \( SL_2(\mathbb{F}_p) \) is equal to \( p+4 \) for an odd prime \( p \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p + 4}
A factory produced $n$ robots. Each robot is defective with probability $q$. If a robot is defective, a tester detects it with probability $p$. We assume that the tester detected $z$ defective robots. Let $X$ be the number of defective robots. Compute $E(X | Z = z)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{z+ (n-z)\frac{(q-q\,p)}{1-q p}}
Suppose we have an infinite double random array $\{X_{ij}:i\geq 1,j\geq 1\}$, where each element is independent and identically distributed with mean zero and variance one. Define $$Y_{i,n}=\frac{1}{n}\sum_{j=1}^nX_{ij}$$ and note that by the strong law of large numbers, for any $i\geq 1$, $$Y_{i,n}\to 0,\quad\text{a.s.}\qquad Y_{i,n}^2\to 0,\quad\text{a.s.}\qquad(n\to\infty).$$ Determine if it is true that $$\frac{1}{n}\sum_{i=1}^nY_{i,n}^2\to 0\quad\text{a.s.}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}