Question

Find the probabilities. \(\frac{C_{4}^{5} C_{0}^{3}}{C_{4}^{8}}\)

Short answer

Question: Explain the issue in the provided exercise related to calculating the probability using the given formula \(\frac{C_{4}^{5} C_{0}^{3}}{C_{4}^{8}}\), and describe the steps to calculate individual combinations. Answer: The issue in the provided exercise is that it requires calculating combinations where the number of items chosen is more than the available items, which is not possible. The given formula has multiple undefined combinations, making the probability calculation impossible. To calculate individual combinations, we use the binomial coefficient formula, \({a \choose b} = \frac{a!}{b!(a-b)!}\), but in this exercise, the combinations \(C_{4}^{5}\) and \(C_{4}^{8}\) cannot be calculated due to the mentioned issue. The exercise seems incorrect or incomplete, and needs corrections before proceeding with the calculation.

Step-by-step solution

Understand the Combinatorial Notation

In the given formula, \(C_{a}^{b}\) represents the number of ways to choose \(b\) items from a set of \(a\) items, also known as "a choose b" or binomial coefficient. We can represent this as \({a \choose b}\). The formula to find the binomial coefficient is: $${a \choose b} = \frac{a!}{b!(a-b)!}$$ Where \(a!\) refers to the factorial of a number, which is the product of all positive integers up to that number.

Calculate the Individual Combinations

Now that we know about combinatory notation, we can calculate the individual combinations given in the formula. 1. \(C_{4}^{5}\): Number of ways to choose 5 items from a set of 4 items. This is not possible as we cannot choose more items than what is available, so \(C_{4}^{5} = 0\). 2. \(C_{0}^{3}\): Number of ways to choose 3 items from an empty set. This would be 0 as well since there are no items to choose from. 3. \(C_{4}^{8}\): Number of ways to choose 8 items from a set of 4 items. This is also not possible due to the same reason mentioned in the first case, so \(C_{4}^{8} = 0\).

Calculate the Probability

Now we need to calculate the probability, given by the formula: $$\frac{C_{4}^{5} C_{0}^{3}}{C_{4}^{8}} = \frac{0 \cdot 0}{0}$$ The above expression is undefined (division by zero). Hence, the probability cannot be determined with the given exercise. It is important in this case to note that the given question seems incorrect or incomplete, as we cannot choose more items than what we have available while calculating combinations. Please check the exercise for any errors and make the necessary corrections before proceeding with the calculation.

Key concepts

Binomial Coefficient

Understanding the binomial coefficient is essential when determining the number of possible combinations in a given scenario. It is represented as \({{a \choose b}}\) or \(C_{a}^{b}\), where it answers the question: how many ways can you choose a group of \(b\) items from a larger set of \(a\) items, without regard to the order?

For instance, if you have a collection of books and want to know how many different sets of 3 books you can arrange from 5 options, you're looking for \(C_{5}^{3}\). To calculate the binomial coefficient, you can use the formula:\[{{a \choose b}} = \frac{a!}{b!(a-b)!}\]Where \(a!\) denotes the factorial of \(a\). It's important to recognize that the binomial coefficient is defined only when \(0 \leq b \leq a\), due to the constraints of choosing no more than the number of available items. If we attempt to choose more, as in \(C_{4}^{5}\), it results in zero, indicating the impossibility of the selection.

Factorial

At the heart of many combinatorial probability calculations lies the factorial, denoted as \(n!\). Factorial represents the product of all positive integers up to number \(n\). Essentially, for any positive integer \(n\), its factorial is given by:\[\(n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1\).\]

It's a concept that extends beyond just whole numbers; for instance, the factorial of zero is defined as \(0! = 1\). This seemingly counterintuitive value is crucial for many equations and probability calculations to hold true. The factorial is fundamental in calculating permutations and combinations, which are the building blocks of probability theory.

Combinations

Combinations involve the selection of items from a larger set where the order of selection does not matter. To illustrate, if you're choosing 2 fruits out of an assortment of 5 (let's say apples, bananas, cherries, dates, and elderberries), it doesn't matter if you pick an apple first and a banana second or a banana first and an apple second; both scenarios represent the same combination (apple and banana).

In math terms, the number of combinations is expressed using the binomial coefficient \({{a \choose b}}\) as mentioned earlier. Key applications of combinations can be found in probability theory when determining the likelihood of an outcome. For example, in a lottery, the combinations of numbers you can pick from a given set can be calculated to understand your chances of winning.

Probability Calculation

Probability calculation is the process of determining the likelihood of a specific event occurring. In formal terms, it is the ratio of the number of favorable outcomes to the total number of possible outcomes.

For instance, if a dice is rolled, the probability of the outcome being a 3 is \(\frac{1}{6}\), since there is only one favorable outcome (rolling a 3) and six possible outcomes in total. When it comes to more complex events, where combinations and permutations come into play, we use the counting principles based on factorials and binomial coefficients to evaluate the probability. In the case of our previous example, the expression \(\frac{C_{4}^{5} C_{0}^{3}}{C_{4}^{8}}\) attempts to calculate such a probability, but it is incorrectly structured because it does not adhere to the rules of combinations wherein the number of items chosen cannot exceed the number available.

Remember, probability always ranges between 0 and 1, inclusive. A probability of 0 means the event is impossible, and a probability of 1 signifies certainty. Interpreting probabilities and determining them accurately is a foundational skill in statistics and various real-world applications, including risk assessment and decision-making.