Question

At a railway station a passenger leaves his luggage in a locker which is opened by dialling a three digit code $($ say $253,009,325$ etc $)$. The passenger chooses the code, close the locker and leaves for the town. A strange man, who does not know the code, tries to open the locker by dialling three digits at random. The probability that the locker opens after $k$ trails is $($ Here $k<1000)$

• A. $\frac{k}{100}$
• B. $\frac{k}{1000}$
• C. $\frac{1000-k}{1000}$
• D. None of these

Answer 1

The correct option is B $\frac{k}{1000}$

$P\left(A\right)=\left(0.1\right)\left(0.1\right)\left(0.1\right)=0.001$
If $k$ trails are made, then it is natural to assume that the unsuccessful combination are not repeated.
Now $P\left(B\right)=1-P\left(\overline{B}\right)$
$=1-\frac{999}{1000}.\frac{998}{999}...\frac{1000-k+1}{1000-k+2}.\frac{1000-k}{1000-k+1}$
$=1-\frac{1000-k}{1000}=\frac{k}{1000}$