A quality control audit has been devised to check on the sampling procedure when a truckload of potatoes arrive at a packing plant. A random sample of 250 is selected and examined for bruises and other defects. The whole truckload will be rejected if 5% of the sample is unsatisfactory. Determine the probability that the shipment will be accepted anyway if the load were to have 8% of the potatoes on the truck not meeting the desired standard.State assumptions and conditions and if they are met.

To determine the probability that the shipment will be accepted, given that 8% of the potatoes on the truck do not meet the desired standard, we need to calculate the probability of having less than 5% unsatisfactory potatoes in a sample of 250.

Assumptions and conditions:
1. The sample of 250 potatoes is randomly selected: This means that each potato in the truckload has an equal chance of being included in the sample, ensuring randomness.
2. The sample is representative of the whole truckload: If the sample is truly random, it should reflect the characteristics of the entire truckload of potatoes.
3. Independent observations: The condition requires that the outcome of one potato being unsatisfactory does not impact the outcome of other potatoes in the sample.

To solve this problem, we can use the binomial distribution formula since the outcome of each potato being satisfactory or unsatisfactory is independent and has only two possibilities.

The binomial distribution formula is:
P(X = k) = (nCk) * p^k * q^(n-k)

Where:
- P(X = k) represents the probability of having exactly k successes in n trials.
- nCk is the binomial coefficient, which is calculated as n! / (k! * (n - k)!)
- p is the probability of success (in this case, the probability of a potato being unsatisfactory)
- q is the probability of failure (the complement of p, q = 1 - p)
- k is the number of successes (in this case, the number of unsatisfactory potatoes in the sample)

In this case, we want to find the probability of having less than 5% (which is 0.05) unsatisfactory potatoes in the sample, given that 8% (0.08) of the potatoes on the truck do not meet the desired standard. So, we need to calculate:

P(X < 0.05 * 250)

Let's calculate the probability using these values:
n = 250 (sample size)
k = 0.05 * 250 = 12.5 (number of unsatisfactory potatoes)

Now, we can calculate the probability using the binomial distribution formula.