Question

Suppose a machine produces metal parts that contain some defective parts with probability $0.05$. How many parts should be produced in order that the probability of at least one part being defective is $\frac{1}{2}$ or more?
(Given ${\log }_{10}95=1.977$ and ${\log }_{10}2=0.3)$

• A. $11$
• B. $12$
• C. $15$
• D. $14$

Answer 1

Answer: (C), (D)

The correct options are
C $15$
D $14$

Probability of at least one part being defective
$=1-$ Probability of no part defective

Probability of a part to be non-defective $=1-0.05=0.95$

Let $n$ be the required number of parts to be produced.
Then, $1-(0.95{)}^n\ge \frac{1}{2}$
$\Rightarrow (0.95{)}^n\le \frac{1}{2}$
$\Rightarrow n({\log }_{10}95-2)\le {\log }_{10}\frac{1}{2}$
$\Rightarrow n\ge \frac{300}{23}$
$n=14,15$