There are 10% defective pens. In a sample of randomly selected 10 pens, the probability that at most one pen is defective, is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (1.9) (0.9)9
The number of defective pens X is binomial with n = 10, p = 0.1, q = 0.9. "At most one" means X = 0 or X = 1:
\( \displaystyle P(X\le 1)=q^{10}+{}^{10}C_{1}\,p\,q^9 \)
\( \displaystyle =(0.9)^{10}+10(0.1)(0.9)^9 \)
Taking out the common factor \( (0.9)^9 \):
\( \displaystyle P(X\le 1)=(0.9)^9(0.9+1) \)
\( \displaystyle =(1.9)(0.9)^9 \)
Why the other options are incorrect
Option (2) (1.1) (0.1)9 — Incorrect. (1.1)(0.1)9 interchanges p and q; the probability of a defective pen is 0.1, so the good pens contribute the powers of 0.9.
Option (3) 0.01 — Incorrect. 0.01 is far too small: P(X = 0) = \( (0.9)^{10}\approx 0.35 \) alone is larger, and \( P(X\le 1)\approx 0.74 \).
Option (4) (0.1) (0.9)9 — Incorrect. (0.1)(0.9)9 is only the term p q9, without the factor \( ^{10}C_{1}=10 \) and without P(X = 0).