GMAT Focus EditionQuantitative ReasoningHard
A production manager is analyzing the defect rate of a manufacturing process. In a sample of 200 items, 10 were found to be defective. If the process is considered to be in control when the defect rate is below 4%, what is the probability that a random sample of 50 items from this process would contain exactly 2 defective items, assuming the observed defect rate holds true?
- A0.224
- B0.205
- C0.182
- D0.246
Show answer & explanationAnswer & explanation
Correct answer: C. 0.182
First, calculate the observed defect rate: 10/200 = 0.05 or 5%. Since this is above the 4% control limit, we use the observed rate for the probability calculation. This is a binomial probability problem with n=50, k=2, and p=0.05. The formula is P(X=k) = C(n, k) * p^k * (1-p)^(n-k).
Why the other options are wrong
- A. This could be a result of using a different k value or a miscalculation in the binomial coefficient.
- B. This might result from a slight miscalculation in combinations or powers, or using an incorrect probability for p.
- D. This would likely result from a significant miscalculation in the binomial probability formula.
Binomial Probability
Binomial probability calculates the likelihood of a specific number of successes in a fixed number of independent trials, given a constant probability of success for each trial.
- Fixed number of trials (n).
- Each trial has only two possible outcomes (success/failure).
- Probability of success (p) is constant for each trial.
- Trials are independent.
Memory trick: Count the chances for exact hits in a set number of tries.