An office supply company ordered a lot of 400 printers. When the lot arrives the company inspector will randomly inspect 12 printers. If more than three printers in the sample are non-conforming, the lot will be rejected. If fewer than two printers are nonconforming, the lot will be accepted. Otherwise, a second sample of size 8 will be taken. Suppose the inspector finds two non-conforming printers in the first sample and two in the second sample. Also AQL and LTPD are 0.05 and 0.10 respectively. Let incoming quality be 4%.

(i) What is the probability of accepting the lot at the first sample?
(ii) What is the probability of accepting the lot at the second sample? 

To solve this problem, we'll use Statistical Quality Control (SQC) principles, specifically acceptance sampling, and probability calculations. We'll also consider the concept of Average Outgoing Quality Limit (AOQL) to determine the probability of accepting the lot at each stage.

First, let's calculate the probabilities for each scenario:

(i) Probability of accepting the lot at the first sample:
The given sample size \( n_1 = 12 \), and the maximum allowable non-conforming printers \( c_1 = 3 \). From the information provided, two non-conforming printers were found in the first sample.

We'll use the binomial distribution formula to calculate the probability:

\[ P(X \leq c_1) = \sum_{x=0}^{c_1} \binom{n_1}{x} p^x (1-p)^{n_1-x} \]

Where:
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