Question

The key for opening a door is in a bunch of $10$ keys. A man attempts to open the door by trying the keys at random discording the wrong key. The probability that the door is opened in the $5^{th}$ trial is

Answer 1

Given there are 10 keys in a bunch

i.e., there are 9 wrong keys and 1 correct key

probability that door is opend in 5th trial is

i.e., in $1^{st},2^{nd},3^{rd}$ and $4^{st}$ trial wrong key is

selected so, probability of selecting wrong key in $1^{st},2^{nd},3^{rd}$

and $4^{st}$ trial is $\frac{9}{10}\times \frac{8}{9}\times \frac{7}{8}\times \frac{6}{7}$

and probability of selecting correct key in $5^{th}$

trial is $\frac{1}{6}$

$\therefore \frac{9}{10}\times \frac{8}{9}\times \frac{7}{8}\times \frac{6}{7}\times \frac{1}{6}=\frac{1}{60}$