This problem can be modeled using the binomial distribution because we have a fixed number of trials (20 components), two possible outcomes for each trial (defective or not defective), and a constant probability of a defective component (5% or 0.05).

Let $X$ be the random variable representing the number of defective components in the sample of 20 components. $X$ follows a binomial distribution: $X \sim \text{Binomial}(n = 20, p = 0.05)$ Where:

• $n = 20$ (sample size),
• $p = 0.05$ (probability of a defective component).

The shipment is accepted if fewer than 2 components are defective, meaning we want to find the probability that $X < 2$.

This can be expressed as: $P(X < 2) = P(X = 0) + P(X = 1)$

Using the binomial probability formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ Where:

• $\binom{n}{k}$ is the binomial coefficient,
• $p^k$ is the probability of having exactly $k$ defective components,
• $(1 - p)^{n - k}$ is the probability of the remaining $n - k$ components being non-defective.

Let's calculate $P(X = 0)$ and $P(X = 1)$.

1. For $X = 0$: $P(X = 0) = \binom{20}{0} (0.05)^0 (0.95)^{20} = 1 \times 1 \times (0.95)^{20}$

2. For $X = 1$: $P(X = 1) = \binom{20}{1} (0.05)^1 (0.95)^{19} = 20 \times 0.05 \times (0.95)^{19}$

Let's now compute the final probability.The probability of accepting a shipment containing 5% defectives, by finding fewer than 2 defective components in the random sample of 20, is approximately 0.736 or 73.6%.

Would you like further details or have any other questions? Here are five related questions you might find helpful:

1. How would the probability change if the sample size increased to 30 components?
2. What happens to the probability if the defective rate increases to 10%?
3. How does the binomial distribution change if we allow up to 3 defective components for acceptance?
4. Can this problem be solved using a Poisson approximation instead of a binomial distribution?
5. How would the company optimize the sample size to minimize the risk of accepting a bad shipment?

Tip: When working with small probabilities and large sample sizes, consider using approximations like the Poisson distribution for efficiency.