14.Probability
medium

एक कक्षा के $60$ विद्यार्थियों में से $30$ ने एन. सी. सी. ( $NCC$ ), $32$ ने एन. एस. एस. $(NSS)$ और $24$ ने दोनों को चुना है। यदि इनमें से एक विद्यार्थी यादृच्छया चुना गया है तो प्रायिकता ज्ञात कीजिए कि

विद्यार्थी ने न तो एन.सी.सी. और न ही एन.एस.एस. को चुना है।

A

$\frac{11}{30}$

B

$\frac{11}{30}$

C

$\frac{11}{30}$

D

$\frac{11}{30}$

Solution

Let $A$ be the event in which the selected student has opted for $NCC$ and $B$ be the event in which the selected student has opted for $NSS$.

Total number of students $=60$

Number of students who have opted for $NCC =30$

$\therefore $ $P(A)=\frac{30}{60}=\frac{1}{2}$

Number of students who have opted for $NSS =32$

$\therefore $ $P(B)=\frac{32}{60}=\frac{8}{15}$

Number of students who have opted for both $NCC$ and $NSS = 24$

$\therefore $ $P ( A$ and $B )=\frac{24}{60}=\frac{2}{5}$

$P ($ not $A$ and not $B)$

$= P(A ^{\prime}$ and  $B ^{\prime})$

$= P \left( A^{\prime} \cap B ^{\prime}\right)$

$= P ( A \cup B )^{\prime}$       $[( A^{\prime} \cap B )=( A \cup B )^{\prime}$  by De Morgan's law $)]$

$=1- P ( A \cup B )$

$=1- P ( A$ or $B )$

$=1-\frac{19}{30}$

$=\frac{11}{30}$

Thus, the probability that the selected students has neither opted for $NCC$ nor $NSS$ is $\frac{11}{30}$

Standard 11
Mathematics

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