**Answer:**

0.0869 = 8.69% probability of getting more than 61% green balls.

**Step-by-step explanation:**

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

**Normal Probability Distribution:**

Problems of normal distributions can be solved using the z-score formula.

In a set with mean and standard deviation , the z-score of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

**Central Limit Theorem**

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean and standard deviation

**The machine is known to have 99 green balls and 78 red balls. **

This means that

**Mean and standard deviation:**

**a. Calculate the probability of getting more than 61% green balls.**

This is 1 subtracted by the pvalue of Z when X = 0.61. So

By the Central Limit Theorem

has a pvalue of 0.9131

1 - 0.9131 = 0.0869

0.0869 = 8.69% probability of getting more than 61% green balls.