To solve this problem, we need to determine the probability that 3 or more parts out of a sample of 50 are defective. This situation involves a binomial probability distribution because:

1. There are two possible outcomes for each part: defective or not defective.
2. The probability of a part being defective is constant ($p = 0.02$).
3. The sample size is fixed ($n = 50$).

Let $X$ represent the number of defective parts in the sample of 50. $X$ follows a binomial distribution: $X \sim \text{Binomial}(n = 50, p = 0.02)$

We are tasked with finding: $P(X \geq 3)$

Step 1: Complement Rule

Using the complement rule, we can write: $P(X \geq 3) = 1 - P(X \leq 2)$

This requires finding $P(X \leq 2)$, which is the cumulative probability of $X = 0, 1,$ or $2$.

Step 2: Formula for Binomial Probability

The probability mass function for a binomial random variable is: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.

Step 3: Calculate $P(X = 0)$, $P(X = 1)$, and $P(X = 2)$

We calculate these probabilities individually:

For $X = 0$:

$P(X = 0) = \binom{50}{0} (0.02)^0 (0.98)^{50} = (1)(1)(0.98^{50}) = 0.3642$

For $X = 1$:

$P(X = 1) = \binom{50}{1} (0.02)^1 (0.98)^{49} = 50 (0.02) (0.98^{49}) = 0.3718$

For $X = 2$:

$P(X = 2) = \binom{50}{2} (0.02)^2 (0.98)^{48} = \frac{50 \cdot 49}{2} (0.02)^2 (0.98^{48}) = 0.1884$

Step 4: Find $P(X \leq 2)$

$P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)$ $P(X \leq 2) = 0.3642 + 0.3718 + 0.1884 = 0.9244$

Step 5: Find $P(X \geq 3)$

Using the complement rule: $P(X \geq 3) = 1 - P(X \leq 2)$ $P(X \geq 3) = 1 - 0.9244 = 0.0756$

Final Answer:

The probability that Bill Dawson finds 3 or more defective parts is approximately: \boxed{0.0756 \, \text{(or 7.56%)}}.

Would you like a detailed explanation of the binomial calculations or graphs of this distribution?

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Here are 5 related questions to expand on this topic:

1. What is the expected number of defective parts in a sample of 50?
2. How would the probability change if the defect rate were 5% instead of 2%?
3. What is the probability of finding exactly 5 defective parts in the sample?
4. How does the shape of the binomial distribution change as the sample size increases?
5. Can this problem be approximated using a normal distribution? If so, how?

Tip: For large sample sizes and small probabilities, consider using the Poisson approximation to simplify binomial probability calculations.