91Ó°ÊÓ

Kombinatorik, or combinatorics, is a branch of mathematics that deals with counting, arrangement, and combination of objects. In this exercise, we use combinatorics to determine how many ways we can select a certain number of people from a group. We used the combination formula \(\binom{n}{k}\), which gives us the number of ways to choose \k\ items from a set of \ items without regard to order. For example, in the step-by-step solution, we calculate the total number of ways to select 4 people out of 10 using this formula: \(\binom{10}{4} = 210\).

Here’s how the combination formula works:

• \(n!\) is the factorial of \, which means the product of all positive integers up to \ (for example, \5! = 5 \times 4 \times 3 \times 2 \times 1\).
• \k!\ is the factorial of \k\.
• \(n - k)!\ is the factorial of \ - k\.

In the context of our problem, \ = 10\ (total people from 5 couples), and \k = 4\ (people to select). Thus, the formula helps us to find all possible groupings.

Ereigniswahrscheinlichkeit, or event probability, is the chance that a specific event will occur out of all possible outcomes. It's a fundamental concept in probability theory and essential for analyzing various scenarios. In this exercise, we want to determine the probability that our selected group of 4 people does not contain any couples.

To find this probability, we:

• First, calculate the total number of possible selections of 4 from 10, which we already did using combinatorics: \(\binom{10}{4} = 210\).
• Then, calculate the number of favorable outcomes, which are selections with no couples. This involves choosing 4 couples out of 5 and then picking one person from each chosen couple. The calculation is: \(\binom{5}{4} = 5\, and for each, \2^4 = 16\, leading to 5 \times\ 16 = 80\).

The probability is then obtained by dividing the number of favorable outcomes by the total number of possible outcomes: \(P = \frac{80}{210} = \frac{8}{21} \).

The probability tells us that if we randomly select 4 people from these 10, there is a \(\frac{8}{21} \) chance that no couples will be among them.

Wahrscheinlichkeitsverteilung, or probability distribution, describes how probabilities are assigned to each possible outcome of a random experiment. It's vital for understanding the full scope of potential results. In our example, we applied the concept by identifying all possible combinations of 4 individuals out of 10 and then determining the distribution of these combinations that meet our criteria—no couples being part of the group.

The steps involved show how the outcomes are distributed:

• Total possible selections: \(\binom{10}{4} = 210\).
• Favorable selections with no couples: \80\.

This exercise isn’t about creating a complete probability distribution, but it gives a snapshot. By summarizing this with a probability fraction, we succinctly characterize part of the distribution: the likelihood of a specific event (no couples) versus all possible selections.

Understanding probability distributions is crucial for more complex analyses, like those involving multiple variables or when assessing different probabilities in larger combinatorial problems.