For independent events A_1, A_2, dots, A_n, P | vedclass

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(C) The probability of non-occurrence of an event $A_i$ is given by $P(A_i^c) = 1 - P(A_i)$.Given $P(A_i) = \frac{1}{i + 1}$, we have $P(A_i^c) = 1 -

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$\frac{1}{n + 1}$

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(C) The probability of non-occurrence of an event $A_i$ is given by $P(A_i^c) = 1 - P(A_i)$.Given $P(A_i) = \frac{1}{i + 1}$, we have $P(A_i^c) = 1 - \frac{1}{i + 1} = \frac{i}{i + 1}$.Since the events $A_1, A_2, \dots, A_n$ are independent, their complements $A_1^c, A_2^c, \dots, A_n^c$ are also independent.The probability that none of the events occur is the product of the probabilities of their non-occurrence:$P(	ext{none occur}) = P(A_1^c) \cdot P(A_2^c) \cdot \dots \cdot P(A_n^c)$$= \left( \frac{1}{2} ight) \cdot \left( \frac{2}{3} ight) \cdot \left( \frac{3}{4} ight) \dots \left( \frac{n}{n + 1} ight)$By canceling the terms in the numerator and denominator, we get:$= \frac{1}{n + 1}$.

For independent events $A_1, A_2, \dots, A_n$, $P(A_i) = \frac{1}{i + 1}$ for $i = 1, 2, \dots, n$. Then the probability that none of the events will occur is:

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