91Ó°ÊÓ

In a binomial distribution, the formula to calculate the probability of exactly \(x\) successes in \(n\) trials is crucial. This formula is: \[ P(X=x) = \binom{n}{x} p^x (1-p)^{n-x} \]

Here, \( \binom{n}{x} \) represents the binomial coefficient. It is calculated as \( \frac{n!}{x!(n-x)!} \). The \(p\) is the probability of success in a single trial, and \( (1-p) \) is the probability of failure. This formula helps you find the probability of getting exactly \(x\) successes out of \(n\) trials.

For example, in the problem, the probability of getting \(x = 3\) successes out of \(n = 12\) trials with \(p = 0.35\) would be calculated like this: \[ P(X = 3) = \binom{12}{3} (0.35)^3 (0.65)^9 \].

This computation involves realizing that the term \(\binom{12}{3}\) is the number of ways to choose 3 successes out of 12 trials.

Cumulative probability refers to the total probability of getting at most a certain number of successes. For the given problem, we need the cumulative probability \(P(X \leq 4)\).

This is calculated by adding up the individual probabilities of getting 0, 1, 2, 3, and 4 successes: \[ P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) \]

Each term in the sum can be computed using the binomial probability formula. In this specific problem, calculating each term individually and then summing them up provides the cumulative probability. When you punch the numbers in, the sum gives you approximately 0.6494.

One of the core assumptions of binomial distribution is that each trial is independent. This means that the outcome of one trial doesn't affect the outcome of another.

For example, if you flip a coin 12 times, the result of each flip doesn't depend on the previous ones.

In our problem, the 12 trials are independent. The probability of success in each trial (0.35) stays constant, regardless of the outcomes of other trials. Independence ensures that the binomial formula works, allowing us to multiply probabilities straightforwardly.

In binomial experiments, the probability of success (\(p\)) in a single trial is a key parameter. In the given problem, the probability of success for each trial is 0.35.

This means that each trial has a 35% chance of success. The probability of failure is then \((1 - p = 0.65)\).

Understanding this parameter helps in applying the binomial probability formula. The probability of success determines the success and failure rates in the formula, making it essential for accurate calculations.

Given \(n = 12\), \(p = 0.35 \), and \( x \leq 4\), the calculations help determine the likelihood of having 0 to 4 successes, leading us to the final cumulative probability of approximately 0.6494.