Question:

$P(X=x)=k(x+1)3^{-x}$, $x=0,1,2,\ldots$ If $k$ is a constant, then $P(X\geq2)$ is equal to

$\dfrac{7}{27}$

$\dfrac{7}{18}$

$\dfrac{11}{18}$

$\dfrac{20}{27}$

Step-by-Step Solution

Key Concept: $\sum_{x=0}^\infty k(x+1)3^{-x}=k\cdot\frac{1}{(1-1/3)^2}=\frac{9k}{4}=1\Rightarrow k=\frac{4}{9}$. $P(X\geq2)=1-P(0)-P(1)$.

$P(X\geq2)=\frac{7}{27}$.

Correct Answer: 1