Judging from recent experience, 5 percent of the worm gears produced by an automatic, high speed machine are defective. What is the probability that out of six gears selected at random, exactly zero gears will be defective?
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A. B. C. D. E.Explanation
Using Binomial probability with n = 6, p = 0.05 and q = 0.95 and r = 0 we get [6!(0.05^0)(0.95^6)]/0!(6 - 0)! = 0.735.
To solve this problem, we can use the binomial probability formula. The formula for the probability of getting exactly 'k' successes in 'n' trials, given a probability 'p' of success in each trial, is:
P(X = k) = (nCk) * (p^k) * ((1 - p)^(n - k))
Where:
In this case, we want to find the probability of selecting exactly zero defective worm gears out of six.
Given that 5 percent of the worm gears are defective, the probability of selecting a defective gear in one trial is 0.05 (or 5% in decimal form). The probability of selecting a non-defective gear is the complement of this, which is 1 - 0.05 = 0.95.
Using the binomial probability formula, we can plug in the values:
P(X = 0) = (6C0) * (0.05^0) * (0.95^(6 - 0)) = (1) * (1) * (0.95^6) = 1 * 1 * 0.735091 ≈ 0.735091
Therefore, the probability that exactly zero gears will be defective is approximately 0.735.
Among the given options, the correct answer is C. 0.735.