To solve this problem, we need to calculate the probability that, in a random sample of 55 tablets from a shipment with a 4% defect rate, we find at most one defective tablet. If that condition is met, the shipment will be accepted.

1. Define Variables:

   • $n = 55$: sample size (number of tablets tested)
   • $p = 0.04$: probability of a tablet being defective
   • $X$: number of defective tablets in the sample

2. Distribution: Since each tablet's defect status is independent of others, we can model $X$, the number of defective tablets in the sample, as a binomial random variable: $X \sim \text{Binomial}(n = 55, p = 0.04)$

3. Probability of Acceptance: The shipment is accepted if there are 0 or 1 defective tablets in the sample. Thus, we need to calculate: $P(X \leq 1) = P(X = 0) + P(X = 1)$

4. Binomial Probability Formula: The probability of $k$ defects in a sample of $n$ tablets, with defect rate $p$, is: $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$ where $\binom{n}{k}$ is the binomial coefficient.

5. Calculate Probabilities:

   • For $P(X = 0)$: $P(X = 0) = \binom{55}{0} (0.04)^0 (0.96)^{55} = (0.96)^{55}$
   • For $P(X = 1)$: $P(X = 1) = \binom{55}{1} (0.04)^1 (0.96)^{54} = 55 \times 0.04 \times (0.96)^{54}$

6. Compute the Values: Now, using the values for $P(X = 0)$ and $P(X = 1)$, we sum them up to find $P(X \leq 1)$.

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To help with this problem, I can walk you through the remaining calculation if you’d like, or you can use the formula and plug in values manually with a calculator. Let me know if you’d like further help on the exact steps.

Would you like details on this, or do you have any other questions? Here are some related questions to consider:

1. How does the binomial distribution apply to quality control sampling?
2. What are alternative ways to calculate probabilities without using binomial formulas?
3. How would changing the sample size from 55 to 100 impact the acceptance probability?
4. What defect rates would lead to near-certain acceptance or rejection?
5. How could you determine a minimum sample size needed to detect defects effectively?

Tip: In quality control, increasing the sample size generally improves the sensitivity of defect detection but may also increase the rejection rate for minor defect levels.