CUET · MATHS · PYQ PAPER 2025

A lot of \(50\) watches is known to have \(10\) defective watches. If \(8\) watches are selected one by one with replacement at random, then the probability that there will be at least one defective watch is:

A \(1-\left(\frac{4}{5}\right)^8\)
B \(1-\left(\frac{4}{5}\right)^{10}\)
C \(1-\left(\frac{1}{5}\right)^2\left(\frac{4}{5}\right)^8\)
D \(1-\left(\frac{1}{5}\right)^2\left(\frac{4}{5}\right)^6\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(1-\left(\frac{4}{5}\right)^8\)

Step-by-step Solution

Detailed explanation

\( P(\text{defective}) = \frac{10}{50} = \frac{1}{5} \) \( P(\text{non-defective}) = \frac{40}{50} = \frac{4}{5} \) \( P(\text{at least one defective}) = 1 - P(\text{no defective in 8 selections}) \) \( P(\text{at least one defective}) = 1 - \left(\frac{4}{5}\right)^8 \)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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