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 एन.एस.एस. को चुना है किंतु एन.सी.सी. को नहीं चुना है।

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 $B )=\frac{24}{60}=\frac{2}{5}$

The given information can be represented by a Venn diagram as

It is clear that Number of students who have opted for $NSS$ but not $NCC$

$=n(B-A)=n(B)-n(A \cap B)=32-24=8$

Thus, the probability that the selected student has opted for $NSS$ but not for $NCC$ 

$=\frac{8}{60}=\frac{2}{15}$