A city has 10 equally populated districts, and a survey is conducted by selecting 100 people from each district. If the probability of a resident owning a bike is 0.2, what is the probability of selecting exactly 150 bike owners from the total survey?

Official Answer

Certainly, navigating through probability questions, especially ones that intricately involve binomial distributions, can be quite enriching. Let me take you through how I would approach this specific problem, drawing upon my experience as a Data Scientist. In essence, we are dealing with a binomial probability question here, where we want to find the probability of having exactly 150 bike owners out of a total of 1000 people surveyed, given that the probability of a resident owning a bike is 0.2.

To start, the total number of people surveyed across the 10 districts is 1000 (since 100 people are selected from each of the 10 districts). We are given that the probability of a resident owning a bike is 0.2. The question at hand asks for the probability of selecting exactly 150 bike owners from these 1000 people. This is a classic example of a binomial distribution scenario, where we have a fixed number of trials (n = 1000), a constant probability of success on each trial (p = 0.2), and we are interested in finding the probability of a certain number of successes (k = 150).

The formula for calculating the probability in a binomial distribution is given by: [P(X = k) = \binom{n}{k} p^k (1-p)^{(n-k)}] Where: - (P(X = k)) is the probability of getting exactly k successes, - (\binom{n}{k}) is the binomial coefficient, which calculates the number of ways to choose k successes from n trials, - (p) is the probability of success on a single trial, - and (1-p) is the probability of failure on a single trial.

Plugging the values into the formula: [P(X = 150) = \binom{1000}{150} (0.2)^{150} (0.8)^{850}] Calculating this directly might seem daunting due to the large numbers involved. However, in a practical scenario, especially within a data science context, we would utilize computational tools and software like Python, specifically leveraging libraries such as scipy which has functions designed to handle binomial distributions efficiently. For example, in Python, you could use:

from scipy.stats import binom
prob = binom.pmf(150, 1000, 0.2)

This approach simplifies the calculation, making it more accessible and less error-prone.

In my professional journey, I've often encountered situations where breaking down complex problems into manageable pieces and utilizing computational tools to address statistical challenges has been key. Whether it was analyzing user behavior data to improve product features, or optimizing algorithms for better performance, the core principle remained the same: leverage your understanding of the problem in conjunction with the right tools to find effective solutions.

This problem, while seemingly straightforward, underscores the importance of a solid grasp on statistical principles combined with computational proficiency. It's a reflection of how, in the field of data science, theoretical knowledge and practical skills come together to solve real-world problems.