In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\left(\displaystyle\frac{9}{10}\right)^5\)
A bulb chosen from the box is defective with probability
\( \displaystyle p = \frac{10}{100} = \frac{1}{10},\quad q = 1 - p = \frac{9}{10} \)
As is usual in such questions, the 5 bulbs of the sample are treated as independent trials with the same p (as when each bulb is put back before the next is drawn). Then the number of defective bulbs X follows a binomial distribution with n = 5 and \(p = \frac{1}{10}\).
\( \displaystyle P(X = 0) = {}^{5}C_{0}\left(\frac{1}{10}\right)^0\left(\frac{9}{10}\right)^5 \)
\( \displaystyle P(X = 0) = \left(\frac{9}{10}\right)^5 \approx 0.590 \)
If the 5 bulbs were drawn together, without replacement, the exact value would be \({}^{90}C_{5} / {}^{100}C_{5} \approx 0.584\). This is very close to 0.590, and no other option is near either value, so the binomial model is the intended one.