• Probability of Non-occurrence of Independent Events:
• Given $n$ mutually independent events $A_1, A_2, \dots, A_n$ with probabilities:

$$P(A_i) = \frac{1}{i + 1}, \quad \text{for } 1 \le i \le n$$

• The complementary probability that event $A_i$ does not occur is:

$$P(A_i^c) = 1 - P(A_i) = 1 - \frac{1}{i + 1} = \frac{i}{i + 1}$$

• By independence of the events, the complements $A_1^c, A_2^c, \dots, A_n^c$ are also independent:

$$P\left(\bigcap_{i=1}^n A_i^c\right) = \prod_{i=1}^n P(A_i^c) = \prod_{i=1}^n \frac{i}{i + 1}$$

• Expanding the telescoping product:

$$P(\text{none occurs}) = \left(\frac{1}{2}\right) \times \left(\frac{2}{3}\right) \times \left(\frac{3}{4}\right) \times \dots \times \left(\frac{n-1}{n}\right) \times \left(\frac{n}{n + 1}\right)$$

• All intermediate numerator and denominator terms cancel out, leaving:

$$P(\text{none occurs}) = \frac{1}{n + 1}$$

• Conclusion: The probability that none of the $n$ events occurs is $\frac{1}{n+1}$ (Option 1).

Correct option: (1) Option 1