Official PYQ · 2005Q52Probability: Random Variables and DistributionsMCQ1 mark

GATE CS 2005 Q52 — Probability: Random Variables and Distributions

A random bit string of length n is constructed by tossing a fair coin n times and setting a bit to 0 or 1 depending on outcomes head and tail, respectively. The probability that two such randomly generated strings are not identical is:

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Answer: D

Let $S_1$ and $S_2$ be two randomly generated bit strings of length $n$ .

Since each bit is generated independently by tossing a fair coin, there are $2^n$ possible unique bit strings of length $n$ , each equally likely to be generated.

For the two strings to be identical ( $S_1 = S_2$ ), whatever string is chosen for $S_1$ , the second string $S_2$ must match it exactly. Since there is only $1$ such matching string out of $2^n$ possibilities:

P(\text{Strings are identical}) = \frac{1}{2^n}

The probability that the two strings are not identical is the complement of this event:

P(\text{Strings are not identical}) = 1 - P(\text{Strings are identical}) = 1 - \frac{1}{2^n}

Therefore, the correct option is D.

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Published 10 Oct 2026

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