Probability Of 6 Workers Taking The Bus A Binomial Distribution Analysis

In the bustling cityscape, where the rhythm of daily commutes shapes the urban landscape, understanding transportation patterns is crucial. One vital aspect of this understanding lies in analyzing how city workers choose to travel to their workplaces. Let's consider a survey indicating that 15% of city workers opt for the bus as their mode of transportation. Now, imagine Donatella, a curious researcher, randomly surveys 10 workers. A fascinating question arises: What is the probability that exactly 6 of these workers take the bus to work? This seemingly simple question delves into the realm of probability distributions, specifically the binomial distribution, which provides a powerful framework for analyzing such scenarios.

Understanding the Binomial Distribution: A Statistical Tool for Success and Failure

To solve this probability problem, we need to understand the concept of the binomial distribution. The binomial distribution is a statistical tool used to model the probability of obtaining a certain number of successes in a sequence of independent trials, where each trial has only two possible outcomes: success or failure. In our case, a "success" would be a worker who takes the bus to work, and a "failure" would be a worker who does not take the bus. The binomial distribution relies on several key assumptions:

1. Fixed number of trials: The experiment consists of a fixed number of trials. In our scenario, Donatella surveys a fixed number of 10 workers, so we have 10 trials.
2. Independent trials: The trials are independent, meaning the outcome of one trial does not affect the outcome of any other trial. We assume that each worker's decision to take the bus is independent of the decisions of the other workers.
3. Two possible outcomes: Each trial has only two possible outcomes: success or failure. As mentioned earlier, a "success" is a worker who takes the bus, and a "failure" is a worker who does not take the bus.
4. Constant probability of success: The probability of success remains constant for each trial. In our case, the probability of a worker taking the bus is 15%, or 0.15, for each worker surveyed.

With these assumptions in mind, we can see that our problem fits the framework of the binomial distribution. The binomial distribution provides a formula to calculate the probability of obtaining exactly k successes in n trials, given the probability of success p on each trial. This formula is given by:

$P(X = k) = {n ext{ choose } k} * p^k * (1 - p)^{(n - k)}$

where:

• $P(X = k)$ is the probability of exactly k successes
• ${n ext{ choose } k}$ is the binomial coefficient, which represents the number of ways to choose k successes from n trials. It is calculated as $n! / (k! * (n - k)!)$, where "!" denotes the factorial function.
• p is the probability of success on a single trial
• n is the number of trials
• k is the number of successes

Applying the Binomial Formula: Calculating the Probability of 6 Bus-Riding Workers

Now that we understand the binomial distribution and its formula, we can apply it to our problem. We want to find the probability that exactly 6 workers out of 10 take the bus to work. So, we have:

• n = 10 (number of workers surveyed)
• k = 6 (number of workers taking the bus)
• p = 0.15 (probability of a worker taking the bus)

Plugging these values into the binomial formula, we get:

$P(X = 6) = {10 ext{ choose } 6} * (0.15)^6 * (1 - 0.15)^{(10 - 6)}$

Let's break down this calculation step by step:

1. Calculate the binomial coefficient:

${10 ext{ choose } 6} = 10! / (6! * 4!) = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1) = 210$

This means there are 210 different ways to choose 6 workers out of 10. 2. Calculate the probability of 6 successes:

$(0.15)^6 = 0.000011390625$

This is the probability of 6 specific workers taking the bus. 3. Calculate the probability of 4 failures:

$(1 - 0.15)^{(10 - 6)} = (0.85)^4 = 0.52200625$

This is the probability of the remaining 4 workers not taking the bus. 4. Multiply the results together:

$P(X = 6) = 210 * 0.000011390625 * 0.52200625 = 0.001249997$

Therefore, the probability that exactly 6 workers out of 10 take the bus to work is approximately 0.00125.

Rounding to the Nearest Thousandth: Presenting the Final Answer

The problem asks us to round the answer to the nearest thousandth. Rounding 0.00125 to the nearest thousandth gives us 0.001.

Therefore, the probability that exactly 6 workers out of 10 take the bus to work is approximately 0.001. This seemingly small probability highlights the fact that observing exactly 6 bus-riding workers in a sample of 10 is a relatively rare event, given that only 15% of the city's workforce uses the bus for their commute. Understanding probabilities like these is crucial for urban planners, transportation officials, and researchers who seek to gain insights into commuting patterns and optimize transportation systems within cities.

Key Takeaways: The Power of the Binomial Distribution

This problem demonstrates the power of the binomial distribution in analyzing scenarios with two possible outcomes. By understanding the assumptions and applying the formula, we can calculate the probability of specific events occurring. In this case, we calculated the probability of exactly 6 workers taking the bus, which turned out to be a relatively small probability. The binomial distribution is a versatile tool with applications in various fields, including statistics, finance, and healthcare. It helps us understand and quantify the likelihood of events, enabling informed decision-making in a world filled with uncertainty. The understanding and application of such statistical tools are very important for a data-driven decision.

In conclusion, the probability of exactly 6 workers taking the bus to work in Donatella's survey is approximately 0.001, highlighting the power of the binomial distribution in analyzing real-world scenarios. This problem illustrates how statistical concepts can be applied to understand and interpret everyday phenomena, making data-driven insights accessible and actionable. As cities continue to evolve and transportation patterns shift, the ability to analyze and interpret probabilities will become increasingly valuable in shaping the future of urban mobility.