(घ) सिद्ध कीजिए कि यदि E और F स्वतन्त्र घटनाएँ हैं, तो E और F' भी स्वतन्त्र होंगी। 5. सभी खण्ड कीजिए- (क) सारणिक 17 15 13 9 -8 7 का मान ज्ञात कीजिए। -3 2 5 (ख) दर्शाइए कि sin-1 12/13 + Cos-1 4/5 + tan-1 63/16 = π. (ग) यद

Step-by-step solution

1. 1

   Given condition

   $E$ and $F$ are independent events.

   $$P(E \cap F) = P(E)P(F)$$
2. 2

   Goal for independence

   To prove $E$ and $F'$ are independent, we must show:

   $$P(E \cap F') = P(E)P(F')$$
3. 3

   Relate $E \cap F'$ to $E \cap F$

   The event $E \cap F'$ means $E$ occurs and $F$ does not occur.

   $$E \cap F' = E - (E \cap F)$$
4. 4

   Probability of $E \cap F'$

   Using the property $P(A-B) = P(A) - P(A \cap B)$:

   $$P(E \cap F') = P(E) - P(E \cap F)$$
5. 5

   Substitute independence

   Substitute $P(E \cap F) = P(E)P(F)$ from the given condition:

   $$P(E \cap F') = P(E) - P(E)P(F)$$
6. 6

   Factor out $P(E)$

   Factor out $P(E)$ from the right side of the equation:

   $$P(E \cap F') = P(E)(1 - P(F))$$
7. 7

   Use complement property

   Recall that $1 - P(F) = P(F')$ (probability of complement event):

   $$P(E \cap F') = P(E)P(F')$$
8. 8

   Conclusion

   Since $P(E \cap F') = P(E)P(F')$, $E$ and $F'$ are independent.