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A keyring contains 'n' keys, with only one being the correct key to unlock a lock. A person attempts to unlock the lock by trying keys randomly, discarding each key after it has been tried. If the probability of successfully unlocking the lock on the kᵗʰ attempt is P, which of the following statements is accurate?
- P is smaller than k/n
- P does not depend on k
- 1/n ≤ P ≤ k/n
- P equals k/n
Correct answer: P does not depend on k
Solution
With keys discarded after each try, P(success on attempt k) = (n-1)/n x (n-2)/(n-1) x ... x 1/(n-k+1) = 1/n, the same for all k. Thus P does not depend on k. The stored 'P smaller than k/n' is not the precise statement; the correct option is that P is independent of k.
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