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STD 12 - 13. probability — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 13. probability1 Mark
MCQ
If the probability that the random variable $X$ takes values $x$ is given by $P ( X = x )= k ( x +1) 3^{- x }, x =0$, $1,2,3 \ldots$, where $k$ is a constant, then $P ( X \geq 2)$ is equal to
  • ✓
    $\frac{7}{27}$
  • B
    $\frac{11}{18}$
  • C
    $\frac{7}{18}$
  • D
    $\frac{20}{27}$
✓

Answer

Correct option: A.
$\frac{7}{27}$
a
$\sum \limits_{x=0}^{\infty} P ( X = x )=1$

$k \left(1+2 \cdot 3^{-1}+3 \cdot 3^{-2}+4 \cdot 3^{-3}+\ldots \infty\right)=1$

$\text { Let } \quad s =1+\frac{2}{3}+\frac{3}{3^2}+\frac{4}{3^3}+\ldots \infty$

$\frac{s}{3}=\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^2}+\ldots \infty$

$\frac{2 s}{3}=1+\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^2}+\ldots \infty$

$\frac{2 s}{3}-\frac{1}{1-\frac{1}{3}}=\frac{3}{2}$

$s=\frac{9}{4}$

$k=\frac{4}{9}$

$P(x \geq 2)=1-P(x=0)-P(x=1)$

$=1-\frac{4}{9}\left(1+\frac{2}{3}\right)$

$=\frac{7}{27}$

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