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The normal approximation is a useful method for estimating probabilities of a binomial distribution, especially when calculating probabilities by hand can become cumbersome. In a binomial experiment where the number of trials is large, the distribution of possible outcomes gets closer to that of a normal distribution. This is known as the Central Limit Theorem, which allows us to use the more manageable normal distribution to approximate binomial probabilities.

To use the normal approximation effectively, certain criteria should be met:

• The number of trials, denoted by \(n\), should be large enough (typically \(n > 30\) is a rule of thumb, but it can work even with \(n\) as low as 20 when \(p\) is not too close to 0 or 1).
• The probability of success, \(p\), should not be too extreme (not too close to 0 or 1).

In the given problem with \(n=20\) and \(p=0.4\), we applied normal approximation by first calculating the mean \(\mu\) and the standard deviation \(\sigma\) of the binomial distribution. These serve as the parameters for conversion to a normal problem, which simplifies the process of finding probabilities.

To calculate the probability using the normal approximation, a sequence of steps helps. Begin by defining the mean and standard deviation of the equivalent normal distribution:
- The mean \(\mu\) is calculated as \(np\).- The standard deviation \(\sigma\) is \(\sqrt{np(1-p)}\).

For this specific problem:

• \(\mu = 20 \times 0.4 = 8\)
• \(\sigma = \sqrt{20 \times 0.4 \times (1-0.4)} \approx 2.19\)

These derived parameters enable the conversion of binomial problems into a framework that utilizes the normal distribution, employing Z-scores.

Next, calculate the Z-score using the formula:\[Z = \frac{x - \mu}{\sigma}\]This standardizes the value, transforming it into a metric that can reference a standard normal distribution table and from which probabilities are accessible.

A particularly important part of using a normal approximation for a discrete binomial distribution is applying the continuity correction. Since the binomial distribution is discrete (distinct whole numbers), but the normal distribution is continuous (a range of values), we introduce a correction to improve approximation accuracy.

This correction involves adjusting our target probability by 0.5 in the direction needed for the inequality:
- To approximate \(P(x \geq k)\), we adjust to \(P(x \geq k - 0.5)\).

In the original example, when finding \(P(x \geq 10)\), you actually calculate \(P(x \geq 9.5)\) using the normal distribution. This shift accounts for the inclusion of values that lie between integer point thresholds, resulting in a more precise prediction.

Utilizing the standard normal table after adjusting for continuity allows us to determine the associated probability more accurately and reflects better alignment with the original discrete scenario.