(a) Write the binomial distribution formula for the probability that 3 comes up exactly 125 times in 720 tosses of a die. (b) Using the normal approximation and tables or calculator, find the approximate answer to (a). (c) Using the normal approximation, find the approximate probability that 3 comes up between 115 and 130 times.

The normal approximation is a useful technique when dealing with binomial distributions, especially for large sample sizes. Instead of working directly with the binomial probabilities, the problem is converted to a normal distribution, which is easier to handle.

To use normal approximation, you need to calculate the mean (μ) and the standard deviation (σ) of the binomial distribution. The mean is found using the formula: \(\text{mean } (\text{μ}) = n \times p\), where:

• \(n\): The number of trials
• \(p\): The probability of success in each trial

For the exercise given, \( n = 720 \) and \( p = \frac{1}{6} \), so the mean (μ) is calculated as \(\text{μ} = 720 \times \frac{1}{6} \).

Next, calculate the standard deviation using the formula: \( \text{σ} = \big\backslash\big\backslashsqrt{n \times p \times (1-p)} \), where the same \( n \) and \( p \) values are used. This way, you end up with a normal distribution that has the calculated mean and standard deviation.

Probability calculation in the context of normal approximation involves finding the area under the normal curve for given intervals. This process often utilizes the Z-score, which transforms a given value into the number of standard deviations it is away from the mean.

To find the probability for a specific range, you first convert the endpoints of that range into Z-scores using the formula: \(\text{Z} = \frac{(X - \text{μ})}{\text{σ}} \), where:

• \(X\): The value from the binomial distribution
• \(\text{μ}\): The calculated mean
• \(\text{σ}\): The calculated standard deviation

For example, to calculate the probability that rolling a 3 appears between 115 and 130 times in 720 tosses, first convert 115 and 130 into Z-scores. Then, find the corresponding probabilities using Z-tables or calculators. This results in approximating the original binomial distribution probability.

A Z-score represents the number of standard deviations a data point is from the mean. It is pivotal in converting binomial distribution problems into the normal distribution format, making calculations simpler.

The Z-score is computed by the formula: \(\text{Z} = \frac{(X - \text{μ})}{\text{σ}} \), where:

• \(X\): The specific value you're evaluating
• \(\text{μ}\): The mean of the distribution
• \(\text{σ}\): The standard deviation of the distribution

For instance, converting the value 125 into a Z-score, given a mean (μ) of 120 and a standard deviation (σ) of approximately 9.46, results in: \(\text{Z} = \frac{(125 - 120)}{9.46} \). Once you have the Z-score, you can use Z-tables to find probabilities related to that score. This helps in determining the probability range for specific values and is crucial for tasks such as calculating the normal approximation of binomial distributions.