Binomial Probability: 6/10 Bus Riders

Jan 12, 2026 by Andrew McMorgan 38 views

Hey guys! Ever wondered about the odds of things happening, especially when you're dealing with a group of people? Well, today we're diving deep into a classic probability problem that's super relevant. Imagine this: a survey found that 15% of city workers take the bus to work. Now, let's say you, Donatella, decide to randomly survey 10 workers. The big question is, what's the probability that exactly 6 of those 10 workers take the bus? This isn't just some random number crunching; understanding this kind of probability helps us make predictions and understand patterns in real-world scenarios, from market research to scientific experiments. We'll be using the binomial probability formula to figure this out, and trust me, by the end of this, you'll be a pro at calculating these kinds of odds. We're going to break down the formula, plug in the numbers, and get you that answer, rounded to the nearest thousandth, so you can impress your friends with your newfound math skills!

Understanding the Binomial Probability Formula

Alright, let's get down to business with the binomial probability formula. This bad boy is your go-to tool whenever you have a situation with a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. In our case, a 'success' is a worker taking the bus, and a 'failure' is a worker not taking the bus. The probability of success remains the same for each trial, which is key here. The formula looks like this: $P(k ext{ successes}) = {}_n C_k imes p^k imes (1-p)^{(n-k)}$ .

Let's break down what each part means:

$n$ : This is the total number of trials. In Donatella's survey, she's talking to 10 workers, so $n = 10$ . This is the size of our sample group.
$k$ : This is the number of 'successes' we're interested in. We want to know the probability that exactly 6 workers take the bus, so $k = 6$ . This is the specific outcome we're aiming for.
$p$ : This is the probability of success on a single trial. The survey tells us that 15% of city workers take the bus, so $p = 0.15$ . This is our baseline probability for any given worker.
$(1-p)$ : This is the probability of failure on a single trial. If $p$ is the probability of taking the bus, then $(1-p)$ is the probability of not taking the bus. So, $(1-p) = 1 - 0.15 = 0.85$ . This is the complement of our success probability.
${}_n C_k$ : This is the binomial coefficient, often read as "n choose k". It represents the number of ways you can choose $k$ successes from $n$ trials, without regard to the order. The formula for this is $\frac{n!}{k!(n-k)!}$ . This part is crucial because it accounts for all the different combinations of workers who might be the bus-takers among the 10 surveyed. For example, it doesn't matter which 6 workers are taking the bus, just that there are exactly 6. This coefficient handles that.

So, to recap, we have our number of trials ( $n=10$ ), our desired number of successes ( $k=6$ ), the probability of success ( $p=0.15$ ), and the probability of failure ( $1-p=0.85$ ). Now, let's plug these numbers into the formula and see what we get!

Calculating the Probability Step-by-Step

Alright, now that we've got the formula and all our variables sorted, it's time to do some actual math, guys! We're going to plug our values into the binomial probability formula step-by-step. Remember, we want to find the probability that exactly 6 out of 10 workers take the bus, given that 15% of all city workers do.

Our formula is: $P(k=6) = {}_{10} C_6 imes (0.15)^6 imes (0.85)^{(10-6)}$ .

Let's tackle each part:

Calculate the binomial coefficient ${}_{10} C_6$ : This tells us how many different ways we can choose 6 workers out of 10. The formula is $\frac{n!}{k!(n-k)!}$ . So, ${}_{10} C_6 = \frac{10!}{6!(10-6)!} = \frac{10!}{6!4!}$ . Let's expand this: $\frac{10 imes 9 imes 8 imes 7 imes 6 imes 5 imes 4 imes 3 imes 2 imes 1}{(6 imes 5 imes 4 imes 3 imes 2 imes 1) imes (4 imes 3 imes 2 imes 1)}$ . We can cancel out the $6!$ from the numerator and denominator: $\frac{10 imes 9 imes 8 imes 7}{4 imes 3 imes 2 imes 1}$ . Now, let's simplify: $\frac{10 imes 9 imes 8 imes 7}{24}$ . $10 imes 9 = 90$ . $8 imes 7 = 56$ . So, $\frac{90 imes 56}{24}$ . $90 imes 56 = 5040$ . $rac{5040}{24} = 210$ . So, there are 210 different ways to choose exactly 6 workers out of 10. Pretty neat, huh?
Calculate $p^k$ : This is the probability of 6 successes. We have $p = 0.15$ and $k = 6$ . So, $(0.15)^6$ . Using a calculator, $(0.15)^6 \approx 0.000011390625$ . This is a very small number, which makes sense because the probability of any single worker taking the bus is already low (15%), and we're looking for 6 out of 10.
Calculate $(1-p)^{(n-k)}$ : This is the probability of the remaining failures. We have $(1-p) = 0.85$ and $(n-k) = (10-6) = 4$ . So, $(0.85)^4$ . Using a calculator, $(0.85)^4 \approx 0.52200625$ . This is the probability that the other 4 workers do not take the bus.
Multiply all the parts together: Now we combine everything: ${}_{10} C_6 imes (0.15)^6 imes (0.85)^4$ . $210 imes 0.000011390625 imes 0.52200625$ . Let's do the multiplication: $210 imes 0.000011390625 \approx 0.00239203125$ . Now, multiply this by $0.52200625$ : $0.00239203125 imes 0.52200625 \approx 0.001248859...$

The Final Answer: Probability Rounded

So, we've gone through all the steps, and our calculation gives us a probability of approximately $0.001248859...$ . The question asks us to round the answer to the nearest thousandth.

Let's look at our number: $0.001248859...$

The thousandths place is the third digit after the decimal point. In our number, that digit is '1'. The digit immediately to its right is '2'. Since '2' is less than 5, we round down, meaning we keep the '1' as it is.

Therefore, rounded to the nearest thousandth, the probability that exactly 6 out of 10 randomly surveyed city workers take the bus is 0.001.

Why is this probability so low?

It's important to pause and think about why this probability is so small. We found that the chance of exactly 6 out of 10 workers taking the bus is only about 0.1%. Remember, the initial survey stated that only 15% of city workers take the bus. This is a fairly low percentage. When you're sampling 10 people, it's much more likely that you'll get a number of bus-takers closer to the average (which would be $10 imes 0.15 = 1.5$ workers). Getting as many as 6 bus-takers out of 10 is quite an outlier event, hence the very low probability. This highlights how binomial probability helps us quantify the likelihood of rare events.

So, next time you're curious about the odds, you know the drill! Use the binomial probability formula, plug in your numbers carefully, and don't forget to round your final answer. It’s a powerful tool for making sense of random chance in the world around us. Keep practicing, guys, and you'll be a probability whiz in no time!