If $A_1, A_2, A_3$ are independent events such that $P(A_i)=\frac{1}{i+1}$, then probability that none occur is:}

Question

From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable $ X $ denote the number of defective items in the sample. If the variance of $ X $ is $ \sigma^2 $, then $ 96\sigma^2 $ is equal to _________.

Answer 1

Concept: For independent events: \[ P(\text{none}) = \prod (1 - P(A_i)) \]

Step 1: {Compute probabilities.}
\[ P(A_1)=\frac{1}{2},\quad P(A_2)=\frac{1}{3},\quad P(A_3)=\frac{1}{4} \]

Step 2: {Compute complements.}
\[ (1-\frac{1}{2})=\frac{1}{2},\quad (1-\frac{1}{3})=\frac{2}{3},\quad (1-\frac{1}{4})=\frac{3}{4} \]

Step 3: {Multiply.}
\[ P = \frac{1}{2}\cdot\frac{2}{3}\cdot\frac{3}{4} = \frac{1}{4} \]

If $A_1, A_2, A_3$ are independent events such that $P(A_i)=\frac{1}{i+1}$, then probability that none occur is:}

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The Correct Option is A

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