Given a binary string of length 10, what is the probability that it contains exactly six 1s if it is known that the string starts with 1?

Instruction: Generate ten independent fair bits and condition on the first bit being 1. Find the probability of exactly six ones in the full string.

Context: This question challenges the candidate to use conditional probability and combinatorial calculations to solve a probability question involving binary strings.

Updated

Official answer

Assume a ten-bit string is chosen uniformly from all 1024 strings, then condition on its first bit being 1. The remaining nine bits are still independent fair bits, so 512 strings remain equally likely.

Exactly six ones overall requires five ones among those nine positions. There are C(9,5)=126 such strings. The conditional probability is 126/512=63/256=0.24609375, about 24.61%.

Do not divide by 1024 after conditioning. If the generator has biased or dependent bits, knowing the first bit is 1 may change the remaining distribution and this count ratio need not apply.

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