91Ó°ÊÓ

Combinatorics is a field of mathematics focused on counting, arranging, and combining objects. It is key to solving problems like the one in our exercise with the bottles. Here, we need to determine how many different ways we can pick two bottles from four. We do this using combinations, which are selections where the order doesn't matter.
To find the number of possible combinations, we apply the formula for combinations:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

where \(n\) is the total number of bottles and \(k\) is the number of bottles we want to pick. For our exercise, \(n = 4\) and \(k = 2\).
Using the formula, we find \(\binom{4}{2} = \frac{4!}{2!(4-2)!} = 6\). This means there are six different combinations possible when selecting two bottles from four: (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), and (3, 4).
All possible combinations are equally likely, which simplifies our probability calculations in later steps.

A random variable in probability is a numerical outcome of a random process. It allows us to quantify scenarios by assigning numbers to different outcomes. In this context, our exercise explores the random variable \(x\), which represents the number of good bottles among two selected ones.
A random variable can take multiple values, depending on the scenario. For example, in our wine bottle selection problem, \(x\) can be:

• 0 if both chosen bottles are bad,
• 1 if one bottle is good and one is bad,
• 2 if both bottles are good.

To determine the value of \(x\) for each outcome, we identify whether the selected bottles are good or bad. By characterizing all possible outcomes from our combinatorial step, we can assess the value of \(x\) for each, paving the way for calculating probabilities.

Calculating the probability of different outcomes involves understanding each event's likelihood and using our understanding of combinatorics and random variables.
In our exercise, there are six equally likely combinations of bottles. The probability of any specific combination, like (1, 2) or (3, 4), is therefore calculated by dividing 1 (certainty of one event happening) by 6 (total number of outcomes), resulting in a probability of \(\frac{1}{6}\) for each.
Next, you calculate the probability distribution for our random variable \(x\) (i.e., number of good bottles):

• \(x = 0\), has one outcome: (1,2), all bad, with probability \(\frac{1}{6}\).
• \(x = 1\), has four outcomes: (1,3), (1,4), (2,3), (2,4), each with one good bottle, with a collective probability of \(\frac{4}{6} = \frac{2}{3}\).
• \(x = 2\), has one outcome: (3,4), both good, with probability \(\frac{1}{6}\).

This makes up the probability distribution of \(x\). By following these calculations, you're able to assess the different probabilities associated with each scenario and understand the likelihood of getting a certain number of good bottles.