An ellipse is inscribed in a circle and a point is chosen at random inside the circle. If the probability that this point lies outside the ellipse is $\frac{2}{3}$,then the eccentricity of the ellipse is $\frac{a\sqrt{b}}{c}$,where $\gcd(a, c) = 1$ and $b$ is a square-free integer. Find the value of $a \cdot b \cdot c$.

$A$ number $n$ is randomly selected from the set $\{1, 2, 3, \dots, 1000\}$. The probability that $\frac{\sum_{i=1}^n i^2}{\sum_{i=1}^n i}$ is an integer is:

Consider the $6 \times 6$ square grid in the figure. Let $A_1, A_2, \ldots, A_{49}$ be the points of intersection (dots in the picture) in some order. We say that $A_i$ and $A_j$ are friends if they are adjacent along a row or along a column. Assume that each point $A_i$ has an equal chance of being chosen.
$(1)$ Let $p_i$ be the probability that a randomly chosen point has $i$ many friends,$i=0, 1, 2, 3, 4$. Let $X$ be a random variable such that for $i=0, 1, 2, 3, 4$,the probability $P(X=i)=p_i$. Then the value of $7 E(X)$ is
$(2)$ Two distinct points are chosen randomly out of the points $A_1, A_2, \ldots, A_{49}$. Let $p$ be the probability that they are friends. Then the value of $7 p$ is

$A$ bag contains $2$ white,$3$ green,and $5$ red balls. If three balls are drawn one after the other without replacement,then the probability that the last ball drawn was red is

If $a$ and $b$ are chosen randomly from the set $\{1, 2, 3, 4, 5, 6\}$ with replacement,then the probability that $\mathop {\lim }\limits_{x \to 0} {\left( {\frac{{{a^x} + {b^x}}}{2}} \right)^{\frac{2}{x}}}=6$ is

$A$ and $B$ are independent events with $P(A)=\frac{1}{4}$ and $P(A \cup B)=2 P(B)-P(A)$,then $P(B)$ is