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MODEL PAPER 2025 — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMODEL PAPER 20253 Marks
Question
A random variable $X$ can take all non negative integral values and the probability that $X$ takes the value is proportional to $5^{-r}$. Find $P(X<3)$
✓

Answer

Given $P(X=r) \alpha \frac{1}{5^r}$
$P(X=r)=k \frac{1}{5^r}\ ($where $k$ is a non$-$zero constant$)$
$P(r=0)=k \cdot \frac{1}{5^0}$
$P(r=1)=k \cdot \frac{1}{5^1}$
$P(r=2)=k \cdot \frac{1}{5^2}$
$P(r=3)=k \cdot \frac{1}{5^3}$
We have, $P(X=0)+P(X=1)+P(X=2)+\ldots \ldots \ldots \ldots=1$
$\Rightarrow k\left(1+\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+\ldots \ldots \ldots\right)=1$
$\Rightarrow k\left(\frac{1}{1-\frac{1}{5}}\right)=1 $
$\Rightarrow k=\frac{4}{5} $
So, $P(X<3)=P(X=0)+P(X=1)+P(X=2)$
$=\frac{4}{5}\left(1+\frac{1}{5}+\frac{1}{5^2}\right)=\frac{4}{5}\left(\frac{25+5+1}{25}\right)=\frac{124}{125} .$

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