Question

The secretary in Exercises 2.121 and 3.16 was given n computer passwords and tries the passwords at random. Exactly one of the passwords permits access to a computer file. Suppose now that the secretary selects a password, tries it, and—if it does not work—puts it back in with the other passwords before randomly selecting the next password to try (not a very clever secretary!). What is the probability that the correct password is found on the sixth try?

Answer 1

Answer:

There is a $\frac{(n-1)^{5}}{n^{6}}$ probability that the correct password is found on the sixth try.

Step-by-step explanation:

We have n passwords, only 1 is correct.

So,

Since a password is put back with other, at each try, we have that:

The probability that a password is correct is $\frac{1}{n}$

The probability that a password is incorrect is $\frac{n-1}{n}$.

There are 6 tries.

We want the first five to be wrong. So each one of the first five tries has a probability of $\frac{n-1}{n}$. So, for the first five tries, the probability of getting the desired outcome is $\frac{(n-1)^{5}}{n^{5}}$.

We want to get it right at the sixth try. The probability of sixth try being correct is $\frac{1}{n}$.

So, the probability that the first five tries are wrong AND the sixth is correct is:

$P = \frac{(n-1)^{5}}{n^{5}}\frac{1}{n} = \frac{(n-1)^{5}}{n^{6}}$

There is a $\frac{(n-1)^{5}}{n^{6}}$ probability that the correct password is found on the sixth try.