An industrial engineer suspects that 35% of all machine malfunctions on the assembly line 
are caused by operator error (not following procedures). The engineer decides to analyze 
accident reports (selected randomly and put back after review) until they find one that 
indicates operator error as the root cause. 1. What is the probability that the engineer will need to review at least three reports before finding one with operator error as the cause?

This problem can be solved using the **Geometric distribution**, which models the number of trials until the first success in a sequence of independent Bernoulli trials (yes/no outcomes). Here, a "success" is defined as finding a report where the cause of the malfunction is operator error.

### Step-by-step approach:
1. **Define the parameters**:
   - The probability of success (finding operator error) is \( p = 0.35 \).
   - The probability of failure (not finding operator error) is \( 1 - p = 0.65 \).
   - The engineer is interested in the probability that it will take **at least three reports** before finding one caused by operator error. 

2. **Geometric distribution**:
   The probability that the first success occurs on the \(k\)-th trial is given by:
   \[
   P(X = k) = (1 - p)^{k-1} \cdot p
   \]
   where \( X \) is the number of trials until the first success.

3. **At least three reports**:
   The phrase "at least three" means we are interested in the probability that the first success occurs on the 3rd trial or later. This can be expressed as:
   \[
   P(X \geq 3) = P(X \geq 3) = 1 - P(X = 1) - P(X = 2)
   \]

4. **Compute individual probabilities**:
   - \( P(X = 1) = (1 - p)^0 \cdot p = 1 \cdot 0.35 = 0.35 \)
   - \( P(X = 2) = (1 - p)^1 \cdot p = 0.65 \cdot 0.35 = 0.2275 \)

5. **Calculate the probability of needing at least three reports**:
   \[
   P(X \geq 3) = 1 - P(X = 1) - P(X = 2)
   \]
   \[
   P(X \geq 3) = 1 - 0.35 - 0.2275 = 1 - 0.5775 = 0.4225
   \]

### Final Answer:
The probability that the engineer will need to review at least three reports before finding one with operator error as the cause is **0.4225**, or **42.25%**.

Would you like further details or have any questions? 

Here are 5 related questions for more practice:
1. What is the probability that the engineer finds a report caused by operator error on the 4th attempt?
2. What is the expected number of reports the engineer will need to review to find one with operator error?
3. How would the probability change if the percentage of malfunctions caused by operator error increased to 50%?
4. What is the probability that the engineer reviews exactly five reports before finding one with operator error?
5. How would this problem change if the reports were not replaced after being reviewed?

**Tip**: The Geometric distribution is useful when modeling "first success" problems, where each trial is independent of the others.

An industrial engineer suspects that 35% of all machine malfunctions on the assembly line are caused by operator error (not following procedures). The engineer decides to analyze accident reports (selected randomly and put back after review) until they find one that indicates operator error as the root cause. What is the probability that the engineer will need to review at least three reports before finding one with operator error as the cause?

Explore the probability problem of an industrial engineer analyzing machine malfunctions caused by operator error. Learn how to apply the Geometric distribution to calculate the chance of needing at least three reports before finding one with operator error. This detailed solution includes key probability formulas and a step-by-step breakdown. Suitable for advanced high school or college students.

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