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?

Official Answer

Certainly, when approaching a probability question like this, especially within the context of my role as a Data Scientist, I find it essential to break down the problem into more manageable components, leveraging both my analytical skills and experience in statistical analysis.

Given we're dealing with a binary string of length 10, and we need to find the probability of this string containing exactly six 1s, knowing it starts with a 1, we're essentially focusing on the remaining 9 positions. Out of these, we need to choose 5 more positions to be 1s to make a total of 6 1s in the entire string. This scenario can be beautifully represented by a combination formula, which is a part of combinatorics, a fundamental concept in probability and statistics that I often use in data modeling and analysis.

To calculate this, we can use the combination formula (C(n, k) = \frac{n!}{k!(n-k)!}), where (n) is the total number of items, and (k) is the number of items to choose. Here, (n=9) because we're excluding the first digit which is already a 1, and (k=5) because we need to choose 5 more 1s. Plugging these values into the formula gives us (C(9, 5) = \frac{9!}{5!(9-5)!}), which simplifies to (\frac{9!}{5!4!}). This equals (\frac{362880}{120 \times 24} = \frac{362880}{2880} = 126).

However, to find the probability, we must consider all possible sequences of the remaining 9 digits, which could either be a 0 or a 1. Thus, there are (2^9) possible sequences. This is the denominator in our probability fraction, representing the total number of outcomes. The numerator, as calculated, is 126, representing the favorable outcomes.

Therefore, the probability is (\frac{126}{2^9}), which simplifies to (\frac{126}{512}). This fraction reduces to approximately (\frac{63}{256}), or in decimal form, about 0.246, which translates to a 24.6% chance. This calculation not only showcases the direct application of combinatorial principles and probability theory but also underscores the importance of a structured analytical approach in solving complex problems.

By dissecting the problem step-by-step, and applying a fundamental statistical formula, we demonstrate not just the technical proficiency expected of a Data Scientist, but also a methodical approach to problem-solving. This technique, adaptable across various data challenges, underscores the value of foundational statistical knowledge coupled with strategic thinking. It's precisely this blend of skills that I've continually honed and which has been pivotal in deriving insights from complex datasets in my career.