Question

A factory $A$ produces $10\%$ defective valves and another factory $B$ produces $20\%$ defective. A bag contains $4$ valves of factory $A$ and $5$ valves of factory $B$. If two valves are drawn at random from the bag. Find the probability that at least one valves is defective. Give your answer upto two places of decimals.

• A. $0.19$
• B. $0.39$
• C. $0.29$
• D. $0.25$

Answer 1

The correct option is C $0.29$

Factory $A^{\prime }$s contribution $=4$ valves ($10\%$ defective)
Factory $B^{\prime }$s contribution $=5$ valves ($20\%$ defective)
Two valves can be selected in ${}^9{C}_2=36$ ways
Now probability ( at least one valve defective) $=1-$P(no defective valve)
The two selected valves may be
A) both from factory $A$
b) both from factory $B$
c) one from each
Therefore required probability $1-\frac{1}{{}^9{C}_2}\left\{{}^4{C}_2{\left(\frac{9}{10}\right)}^2{+}^5{C}_2{\left(\frac{4}{5}\right)}^2{+}^4{C_1}^5{C}_1\frac{9.4}{10.5}\right\}$
$=1-\frac{1}{36}\{\frac{486}{100}+\frac{160}{25}+\frac{720}{50}\}=1-\frac{2566}{3600}=\frac{517}{1800}$