Bus Commute Probability: A City Worker Survey

by Andrew McMorgan 46 views

Hey guys! Ever wondered about the odds of randomly picking a group of people and finding out how many of them take the bus to work? Well, let's dive into a fun little probability problem that's super relevant to city life. Imagine we're doing a survey in a city where 15% of all workers use the bus to get to their jobs. Now, if we randomly ask 10 workers, what's the chance that exactly 6 of them are bus riders? This isn't just a theoretical head-scratcher; it’s the kind of question that helps urban planners and transit authorities understand commuting patterns and make better decisions about public transportation. So, buckle up as we break down this probability puzzle step by step!

Understanding the Problem

When tackling probability questions like this, it's crucial to first understand the type of probability distribution we're dealing with. In this case, we're looking at a binomial distribution. Why binomial? Because we have a fixed number of trials (10 workers surveyed), each trial is independent (one worker's choice doesn't affect another's), there are only two possible outcomes (either a worker takes the bus or they don't), and the probability of success (a worker taking the bus) is constant (15% or 0.15). Identifying these characteristics helps us choose the right formula and approach.

Now, let's define our terms. We have 'n', which is the number of trials (10 workers). We want to find the probability of 'k' successes (exactly 6 workers taking the bus). The probability of success on a single trial 'p' is 0.15 (the percentage of workers who take the bus). And the probability of failure 'q' is 1 - p, which is 0.85 (the percentage of workers who don't take the bus). Understanding these variables is the first step to cracking this problem. Once we have a clear grasp of what each variable represents, we can plug them into the binomial probability formula and calculate the exact probability we're looking for. This careful setup ensures that we're not just blindly applying a formula but actually understanding the underlying concepts.

The Binomial Probability Formula

The binomial probability formula is the key to solving this problem. It's expressed as:

P(X=k)=nchoosekβˆ—pkβˆ—q(nβˆ’k)P(X = k) = {n choose k} * p^k * q^(n-k)

Where:

  • P(X=k)P(X = k) is the probability of getting exactly k successes in n trials.
  • nchoosek{n choose k} is the number of combinations of n items taken k at a time, also known as the binomial coefficient.
  • pp is the probability of success on a single trial.
  • qq is the probability of failure on a single trial (which is 1βˆ’p1 - p).

Let's break this down even further. The term nchoosek{n choose k} might look intimidating, but it's just a way of calculating how many different ways you can choose k successes out of n trials. For example, if you're picking 6 workers out of 10, it tells you how many different groups of 6 you could possibly form. The formula for nchoosek{n choose k} is:

{n choose k} = RAC{n!}{k!(n-k)!}

Where n! (n factorial) means n Γ— (n-1) Γ— (n-2) Γ— ... Γ— 2 Γ— 1. So, 5! = 5 Γ— 4 Γ— 3 Γ— 2 Γ— 1 = 120. Understanding this formula is crucial because it accounts for all the different ways you can achieve the exact number of successes you're interested in. Without it, you'd only be calculating the probability of one specific sequence of successes and failures, not the overall probability of getting k successes in any order.

Applying the Formula to Our Problem

Alright, let's get our hands dirty and apply the binomial probability formula to our city worker problem. We know that:

  • n = 10 (number of workers surveyed)
  • k = 6 (number of workers who take the bus)
  • p = 0.15 (probability of a worker taking the bus)
  • q = 0.85 (probability of a worker not taking the bus)

First, we need to calculate the binomial coefficient 10choose6{10 choose 6}:

{10 choose 6} = RAC{10!}{6!(10-6)!} = RAC{10!}{6!4!} = RAC{10 * 9 * 8 * 7 * 6!}{6! * 4 * 3 * 2 * 1} = RAC{10 * 9 * 8 * 7}{4 * 3 * 2 * 1} = 210

So, there are 210 different ways to choose 6 workers out of 10. Now we plug everything into the binomial probability formula:

P(X=6)=10choose6βˆ—(0.15)6βˆ—(0.85)4P(X = 6) = {10 choose 6} * (0.15)^6 * (0.85)^4

P(X=6)=210βˆ—(0.15)6βˆ—(0.85)4P(X = 6) = 210 * (0.15)^6 * (0.85)^4

Now, let's calculate (0.15)6(0.15)^6 and (0.85)4(0.85)^4:

(0.15)6approx0.00001139(0.15)^6 approx 0.00001139

(0.85)4approx0.5220(0.85)^4 approx 0.5220

Finally, multiply all the terms together:

P(X=6)=210βˆ—0.00001139βˆ—0.5220approx0.00125P(X = 6) = 210 * 0.00001139 * 0.5220 approx 0.00125

So, the probability that exactly 6 out of 10 randomly surveyed city workers take the bus to work is approximately 0.00125.

Rounding to the Nearest Thousandth

The question asks us to round the answer to the nearest thousandth. Our calculated probability is approximately 0.00125. To round this to the nearest thousandth, we look at the digit in the ten-thousandths place (the fourth digit after the decimal point), which is 2. Since 2 is less than 5, we round down, meaning we keep the thousandths digit as it is.

Therefore, the probability rounded to the nearest thousandth is 0.001. This means that if you randomly survey 10 city workers, there's only about a 0.1% chance that exactly 6 of them will be bus commuters. It's a pretty small probability, highlighting that while 15% of all city workers take the bus, finding a group of 10 where more than half are bus riders is quite rare.

Conclusion

So, there you have it! The probability that exactly 6 out of 10 randomly surveyed city workers take the bus to work is approximately 0.001 when rounded to the nearest thousandth. This problem showcases how the binomial probability formula can be used to solve real-world scenarios. Whether you're planning transportation, conducting surveys, or just curious about the odds, understanding these concepts can be incredibly useful. Keep exploring the world of probability, guys, because you never know when these skills might come in handy!