Probability: 1 Woman & 2 Men Committee

Let's dive into a classic probability problem! We're going to figure out the chances of forming a committee with a specific gender mix. In this case, we have a classroom with a group of bright individuals, and we need to select a small team from them. Specifically, we have 8 women and 5 men, and the goal is to form a committee of 3 people. What we want to know is: what's the probability that our committee ends up with 1 woman and 2 men?

Understanding the Basics of Probability and Combinations

Before we jump into the calculations, let's quickly recap the core concepts we'll be using. Probability, at its heart, is about figuring out how likely an event is to occur. It's often expressed as a fraction: the number of favorable outcomes divided by the total number of possible outcomes. So, in our case, the favorable outcome is selecting a committee with 1 woman and 2 men, and the total possible outcomes are all the different 3-person committees we could form.

Now, this is where combinations come into play. A combination is a way of selecting items from a set where the order doesn't matter. Think of it like this: if we're picking people for a committee, it doesn't matter if we pick Alice then Bob then Carol, or Carol then Bob then Alice – it's the same committee. The formula for combinations is written as "n choose k" or C(n, k), where 'n' is the total number of items and 'k' is the number of items we're choosing. The formula itself looks like this:

C(n, k) = n! / (k! * (n-k)!)

Where "!" represents the factorial (e.g., 5! = 5 * 4 * 3 * 2 * 1). This formula might look intimidating, but it's really just a way to count the number of different ways we can pick a group of items without worrying about the order. For example, if we want to choose 2 people out of 5, we'd calculate C(5, 2) = 5! / (2! * 3!) = 10. This means there are 10 different ways to pick a group of 2 people from a group of 5.

Why are combinations so crucial here? Because when we're forming our committee, the order in which we select the members is irrelevant. We care about the final composition of the committee – who's on it – not the sequence in which they were chosen. This is why the combination formula becomes our go-to tool for calculating the number of possible committees and the number of committees with our desired gender mix. Ignoring combinations and simply multiplying probabilities would lead to a drastically incorrect answer, as it would not account for the overlapping possibilities and the fact that the order of selection doesn't change the final committee makeup. Understanding this distinction is fundamental to tackling probability problems involving selections from a group.

Calculating the Number of Favorable Outcomes

Okay, with the basics down, let's get to the heart of the problem. We need to figure out how many ways we can form a committee with 1 woman and 2 men. This is where we break the problem down into smaller, manageable steps, using our newfound knowledge of combinations.

First, let's think about the women. We need to choose 1 woman out of the 8 available. This is a combination problem, so we use the formula: C(8, 1). Plugging the numbers in, we get:

C(8, 1) = 8! / (1! * 7!) = 8

So, there are 8 different ways to choose 1 woman from the group of 8. Now, let's think about the men. We need to choose 2 men out of the 5 available. Again, this is a combination, so we use C(5, 2):

C(5, 2) = 5! / (2! * 3!) = 10

There are 10 different ways to choose 2 men from the group of 5. Here's where the magic happens: to get the total number of committees with 1 woman and 2 men, we multiply these two results together. Why do we multiply? Because for each way we can choose a woman, there are multiple ways we can choose the men. It's like a tree diagram – each branch for the women selection splits into multiple branches for the men selection. So, the total number of favorable outcomes is:

8 * 10 = 80

This means there are 80 different possible committees that consist of exactly 1 woman and 2 men. It's crucial to understand why we multiply here. Each of the 8 ways to choose a woman can be combined with each of the 10 ways to choose the men. If we were to add them, it would suggest these selections are mutually exclusive, which they are not. The act of selecting the woman and the men are independent events that jointly contribute to the composition of the committee. This multiplication principle is fundamental in combinatorial problems where you need to count the outcomes of multiple independent selections or events happening together. By breaking the problem into smaller combination calculations and then multiplying the results, we systematically arrive at the total number of favorable outcomes.

Calculating the Total Number of Possible Outcomes

We've figured out the number of ways to get our desired committee composition (1 woman and 2 men). Now, we need to know the total number of possible committees we could form, regardless of gender. This will be our denominator in the probability fraction.

Remember, we're forming a committee of 3 people from a total pool of 13 people (8 women + 5 men). So, this is another combination problem: we're choosing 3 people out of 13, without regard to order. We use the combination formula C(13, 3):

C(13, 3) = 13! / (3! * 10!) = (13 * 12 * 11) / (3 * 2 * 1) = 286

So, there are 286 different possible committees we could form. This number represents the entire sample space – all the possible outcomes when we select a committee of 3 from the combined group of women and men. It's crucial to calculate this accurately because it forms the basis against which we compare our favorable outcomes (the 1 woman and 2 men committees). The total possible outcomes account for every possible group of three that could be selected, irrespective of gender balance. This includes committees with all women, all men, or other mixed combinations. Understanding the total number of possibilities provides the necessary context for assessing how likely our specific desired outcome (1 woman, 2 men) is compared to all other possibilities. This step sets the stage for the final calculation of the probability, where we'll express the favorable outcomes as a fraction of this total.

Calculating the Probability

Alright, we've done the hard work! We know the number of favorable outcomes (80 committees with 1 woman and 2 men) and the total number of possible outcomes (286 committees). Now, we just need to put it all together to calculate the probability.

Remember, probability is defined as:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

In our case, this translates to:

Probability = 80 / 286

Now, we can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2:

Probability = 40 / 143

So, the probability of forming a committee with 1 woman and 2 men is 40/143. This is our final answer! To put it in perspective, this fraction represents the proportion of all possible 3-person committees that would consist of 1 woman and 2 men, given the specific gender distribution of our class. A probability of 40/143 suggests that this particular committee composition is certainly possible, but not the most likely outcome. There are other combinations of committee members that are more or less likely, and this specific probability quantifies the chances of hitting this particular mix.

To make this probability even more relatable, we could express it as a percentage. Dividing 40 by 143 gives us approximately 0.2797, which translates to about 27.97%. This means that if we were to randomly form a large number of 3-person committees from this class, we'd expect roughly 28% of them to have 1 woman and 2 men. This percentage provides an intuitive sense of how frequently this specific outcome would occur in practice.

Conclusion

So, guys, we've successfully tackled a probability problem involving combinations! We figured out the probability of forming a committee with 1 woman and 2 men from a class of 8 women and 5 men. We covered the basics of probability and combinations, calculated the number of favorable and total outcomes, and finally arrived at the answer: a probability of 40/143, or approximately 27.97%. This problem highlights how breaking down complex scenarios into smaller steps and using the right tools (like the combination formula) can make even seemingly daunting calculations manageable. Keep practicing, and you'll be a probability pro in no time!