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Probability theory is the branch of mathematics that studies randomness and uncertainty. It's the science of quantifying the likelihood of different outcomes occurring. In our problem, we are dealing with a scenario where workers may or may not take public transportation on a given day. This situation can be modeled using a binomial distribution.

The key concept here is the probability of a success, which is when a worker takes public transportation. We are given that this probability is 30%, or 0.3. In probability theory, this is denoted as \( p = 0.3 \). Each worker taking public transportation is a random experiment with two possible outcomes: they either take it or they don't.

A binomial distribution is appropriate when you have a fixed number of independent experiments, each with the same probability of success. Here, we have 10 trials, or experiments, with each worker's decision to take public transportation being independent of the others. The binomial formula gives us a way to compute the probability of getting exactly \( k \) successes (i.e., workers taking public transport) out of \( n \) trials.

Statistical analysis involves using mathematical strategies to comprehend data. In this task, we perform statistical analysis through the use of the binomial probability formula. This allows us to investigate how many workers, in a sample, might regularly take public transport.

The analysis requires calculating probabilities for specific outcomes. For Part (a), we find the probability that exactly 3 workers take public transport by using the formula:
\[ P(X = 3) = \binom{n}{k} p^k q^{n-k} \]
where \( \binom{n}{k} \) signifies the combinations and \( p \) and \( q \) represent the probabilities of success and failure, respectively. Breaking it down into steps simplifies the process and ensures accuracy in calculation.

For Part (b), the approach involves finding the probability that at least 3 workers take public transport. This sub-task involves cumulative probability, where we first compute the probability for fewer than 3 workers and then subtract from 1 to find the complementary probability for 3 or more workers. Statistical analysis helps us draw meaningful conclusions from these probabilistic calculations.

Quantitative methods involve mathematical and statistical tools to solve problems and make decisions. They enable us to handle data and derive results that can inform real-world decisions. In this exercise, quantitative methods are represented by the use of the binomial distribution and the application of binomial coefficients.

To tackle Part (a), we employed the binomial coefficient, \( \binom{10}{3} \), which calculates the number of ways to choose 3 successes out of 10 trials. This coefficient is vital because it allows us to see how combinations of successes and failures can occur in different sequences.

Additionally, quantitative methods help in simplifying complex calculations. For example, in Part (b), using the complement rule \( 1 - P(X < 3) \) made calculating the probability of at least 3 workers much simpler. The complement rule is a common quantitative technique that provides efficiencies in problems involving distributions.

Understanding these methods enhances comprehension of probability-related questions and boosts one’s ability to perform statistical analyses effectively.