Three independent events. `A_(1), A_(2)` and `A_(3)` occur with probabilities `P(A_(i))= 1/(1+i) ,i=1, 2, 3`. What is the probability that at least one of the three events occurs?

Options

A. `(1)/(4)`
B. `(2)/(3)`
C. `(3)/(4)`
D. `(1)/(24)`

Answer

To find the probability that at least one of the three independent events \( A_1, A_2, \) and \( A_3 \) occurs, we can follow these steps:

### Step 1: Determine the probabilities of each event
Given the probabilities:
- \( P(A_1) = \frac{1}{1+1} = \frac{1}{2} \)
- \( P(A_2) = \frac{1}{1+2} = \frac{1}{3} \)
- \( P(A_3) = \frac{1}{1+3} = \frac{1}{4} \)

### Step 2: Calculate the probabilities of the complements of each event
The complements of the events 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 none of the events occur
Since the events are independent, the probability that none of the events occur is given by:
\[ P(A_1' \cap A_2' \cap A_3') = P(A_1') \times P(A_2') \times P(A_3') \]
Calculating this:
\[ P(A_1' \cap A_2' \cap A_3') = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \]
\[ = \frac{1 \times 2 \times 3}{2 \times 3 \times 4} = \frac{6}{24} = \frac{1}{4} \]

### Step 4: Calculate the probability that at least one event occurs
The probability that at least one of the events occurs is:
\[ P(A_1 \cup A_2 \cup A_3) = 1 - P(A_1' \cap A_2' \cap A_3') \]
Substituting the value we found:
\[ P(A_1 \cup A_2 \cup A_3) = 1 - \frac{1}{4} = \frac{3}{4} \]

### Final Answer
Thus, the probability that at least one of the three events occurs is \( \frac{3}{4} \).

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