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In a binomial distribution, calculating probability involves a specific formula. This formula considers both the total number of trials and the number of successful outcomes. When calculating the probability of a particular number of successes in a binomial distribution, like in our example, we use:
\[P(x=k) = C(n, k) \times (p^k) \times (1-p)^{n-k}\]
Here,

• \(P(x=k)\) is the probability of observing exactly \(k\) successes in \(n\) trials.
• \(C(n, k)\) is the combination, or number of ways to choose \(k\) successes from \(n\) trials.
• \(p\) is the probability of success on a single trial.
• \((1-p)\) is the probability of failure on a single trial.

This formula incorporates both the likelihood of each success and failure and the number of ways they can occur together. To find the probability of 2 households having cable TV when selected from 4, simply plug the appropriate values into the formula.

A random variable is a key concept in probability theory. It represents a numerical value that describes the outcome of a random process. In the context of the binomial distribution, a random variable often represents the number of successful outcomes in a series of trials.
Let's think of the example given: the number of households with cable TV among four randomly selected households. Here, the random variable \(x\) can take values between 0 and 4, corresponding to the number of households with cable TV.
Understanding this variable helps in identifying the range of possible outcomes and assigning probabilities using the binomial probability formula. By considering each possible value of \(x\) separately, we can compute precise probabilities for each scenario, aiding in deeper analysis of real-world situations.

Combinatorial probability plays a vital role in determining the probability of various events within the binomial model. It refers to choosing specific outcomes from a larger set of possibilities, which is often achieved using combinations.
In our problem, combinations help us figure out how many ways we can select a certain number of households with cable TV from the total. The formula for combinations is given by:
\[C(n, k) = \frac{n!}{k!(n-k)!}\]

• \(n!\) (read as "n factorial") is the product of all positive integers up to \(n\).
• \(k!\) and \((n-k)!\) are the factorials of \(k\) and \(n-k\), respectively.

In essence, combinations allow us to calculate the number of different ways a particular outcome can occur, helping to accurately assess its probability. This makes it essential in solving any problem involving multiple trials and the binomial distribution.