SmartLearn employs a team of 2 helpdesk operators per shift who, together, can resolve a maximum of 20 support tickets (requests) per hour. 

What is the probability that the team is unable to handle all incoming support tickets (requests) during a one-hour period? 

a) Clearly define the random variable and the probability expression.

b) Present your numeric result and provide a brief recommendation to SmartLearn on whether their current staffing is likely to meet demand, or if adjustments may be necessary.

c) Would a normal approximation be appropriate in this case? Justify your answer based on distribution properties.

What is the most appropriate probability distribution for modeling X? 

Statistical models generally rely on a set of assumptions. Identify a key assumption behind the model for X, and explain how it might be violated in the SmartLearn context. Provide a realistic example supporting your argument.

Suppose SmartLearn changes its automated review system criteria, which results in an increase of the automated approval rate by 10 percentage points. 

How would this change affect the expected value and variability of the number of video submissions attempts until automated approval?

Based on your results above, is Sarah's experience common or unusual? Briefly explain your reasoning using the probabilities from parts (a) and (b) from question (11) .

Every video submission at SmartLearn incurs a base cost of $0.25 for automated review. If the video is automatically approved, it proceeds to manual review, which adds $1.50 to the cost.

a) Write down the formulas for the expected value and variance of review cost per video submission as a function of the random variable auto_approved. 

b) Compute the expected review cost per video submission.

c) Compute the variance of the review cost per video submission.

Note: If you were unable to estimate the probability in the previous question, assume that the probability of automatic approval is 0.76 for this question.

Using the available information, estimate the probability that a video is automatically approved on any given attempt. 

Use "." as a decimal separator and present your result with 2 decimal places.

In Table 3, the column titled "N" displays three distinct values across variables.  What does 'N' represent and why does it take on different values for different variables? Describe what each of the three values corresponds to in terms of the data.

What is the most appropriate probability distribution for modeling auto_approved?