In a box consisting 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\left(\displaystyle\frac{9}{10}\right)^5\)
Of the 100 bulbs, 90 are not defective. For any one bulb drawn:
\( \displaystyle P(\text{not defective}) = \frac{90}{100} = \frac{9}{10} \)
As usual in such questions, the 5 bulbs of the sample are treated as drawn one at a time with replacement, so the five draws are independent events and the probabilities multiply:
\( \displaystyle P(\text{none defective}) = \frac{9}{10} \times \frac{9}{10} \times \frac{9}{10} \times \frac{9}{10} \times \frac{9}{10} = \left(\frac{9}{10}\right)^5 \)
Why the other options are incorrect
Option (1) 10−1 — Incorrect. 1/10 is the probability that a single bulb is defective.
Option (2) \(\left(\displaystyle\frac{1}{2}\right)^5\) — Incorrect. It wrongly treats defective and non-defective as equally likely.
Option (4) \(\left(\displaystyle\frac{9}{10}\right)\) — Incorrect. This is the probability for one bulb only; all five must be non-defective.