ProbabilityHardBloom L3
Question
Two integers $x$ and $y$ are chosen from the set $\{ 0,1,2,3,\ldots,2n - 1,2n\}$, with replacement. The probability that $|x - y| \leq n,n \in N$ is
Options
A.$\frac{3n^{2} + 1}{(2n + 1)^{2}}$
B.$\frac{3n^{2} + 3n - 1}{(2n + 1)^{2}}$
C.$\frac{3n^{2} + 3n + 1}{(2n + 1)^{2}}$
D.$\frac{3n(n + 1)}{(2n + 1)^{2}}$
Solution
Sol. [Key Concept: Integers $x,y$ from $\{ 0,1,\ldots\}$ under the condition: count pairs.]
$(x - y) = n\ $ number of $(x,y) = n + 1$
$$\begin{matrix} = n - 1 & \text{~}\text{number of}\text{~}(x,y) = n + 2 \\ \vdots & \\ = 1 & = 2n \\ = 0 & = 2n + 1 \end{matrix}$$
∴ Number of favorable ways $= 2((n + 1) + (n + 2) + (n + 3) + \ldots(2n)) + 2n + 1$
$$\begin{matrix} & \ = 3n^{2} + 3n + 1 \\ P & \ = \frac{3n^{2} + 3n + 1}{(2n + 1)^{2}} \end{matrix}$$
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