Let `P(A_i),` the probability of happening of independent events `A_i(i=1, 2, 3) " be given by " P(A_i)=1/(i+1)`
Statement-1 : The probability that at least one event happens is `3/4`.
. Statement-2 : `P(uuu_(i=1)^(3)Ai)=1-prod_(i=1)^(3)P(A'_(i))`

Options

A. Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement -1
B. Statement-1 is true, Statement-2 is true, Statement-2 is Not a correct explanation for statement -1
C. Statement-1 is true, Statement-2 is False
D. Statement-1 is False, Statement-2 is true

Answer

To solve the problem, we will analyze the given statements and calculate the required probabilities step by step.

### Step 1: Calculate the probabilities of the events \( A_1, A_2, A_3 \)

Given:
- \( P(A_i) = \frac{1}{i+1} \)

Calculating for \( i = 1, 2, 3 \):
- \( P(A_1) = \frac{1}{1+1} = \frac{1}{2} \)
- \( P(A_2) = \frac{1}{2+1} = \frac{1}{3} \)
- \( P(A_3) = \frac{1}{3+1} = \frac{1}{4} \)

### Step 2: Calculate the probabilities of the complements of the events

The complements are:
- \( P(A_1') = 1 - P(A_1) = 1 - \frac{1}{2} = \frac{1}{2} \)
- \( P(A_2') = 1 - P(A_2) = 1 - \frac{1}{3} = \frac{2}{3} \)
- \( P(A_3') = 1 - P(A_3) = 1 - \frac{1}{4} = \frac{3}{4} \)

### Step 3: Calculate the probability that at least one event happens

Using the formula for the probability of the union of independent events:
\[ P(A_1 \cup A_2 \cup A_3) = 1 - P(A_1') \cdot P(A_2') \cdot P(A_3') \]

Substituting the values:
\[ P(A_1 \cup A_2 \cup A_3) = 1 - \left(\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4}\right) \]

Calculating the product:
\[ P(A_1') \cdot P(A_2') \cdot P(A_3') = \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} = \frac{1 \cdot 2 \cdot 3}{2 \cdot 3 \cdot 4} = \frac{6}{24} = \frac{1}{4} \]

Now substituting back:
\[ P(A_1 \cup A_2 \cup A_3) = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 4: Verify Statement 1

Statement 1 claims that the probability that at least one event happens is \( \frac{3}{4} \). We have calculated and confirmed that:
\[ P(A_1 \cup A_2 \cup A_3) = \frac{3}{4} \]
Thus, Statement 1 is **true**.

### Step 5: Verify Statement 2

Statement 2 states:
\[ P\left(\bigcup_{i=1}^{3} A_i\right) = 1 - \prod_{i=1}^{3} P(A_i') \]

We already calculated:
\[ P\left(\bigcup_{i=1}^{3} A_i\right) = \frac{3}{4} \]
And we calculated:
\[ \prod_{i=1}^{3} P(A_i') = \frac{1}{4} \]

Thus:
\[ 1 - \prod_{i=1}^{3} P(A_i') = 1 - \frac{1}{4} = \frac{3}{4} \]

This confirms that Statement 2 is also **true**.

### Conclusion

Both statements are true.