Question

A spyware is trying to break into a system by guessing its password. It does not give up until it tries 1 million different passwords. What is the probability that it will guess the password and break in if by rules, the password must consist of
(a) 6 different lower-case letters
(b) 6 different letters, some may be upper-case, and it is case-sensitive
(c) any 6 letters, upper- or lower-case, and it is case-sensitive
(d) any 6 characters including letters and digits

Answer 1

Answer: So our limit is 1 million attempts.

a) 6 lower case letters.

Think in 6 blank spaces where you can put one letter, so in each space you have 26 options, so the total of combinations is $26^{6}$ = 308915776.

so if you have 1 million attempts, then 1000000/308915776 = 0.00323

so you have 0.32% chances of breaking the code.

b) 6 different letters, some may be upper-case, and it is case-sensitive

Again 6 blank spaces, in the first you can put 26 letters,  and because its case sensitive, you have 52 options, in the second space, you will have 25 posible letters, and 50 options, and so on.

So the total combinations are 52*50*48*46*44*42 = 10608998400

and the probability of breaking it with 1 mill attempts is = 0.009%

(c) any 6 letters, upper- or lower-case, and it is case-sensitive

same as the first case, but here you have 26 lower case and 26 upper, so you have $52^{6}$ = 19770609664 combinations

and in 1 mill attempts the probability of breaking it is 0.0005%

(d) any 6 characters including letters and digits

Here you have 26 letters and 10 numbers, so the total combinations are :

$36^{6}$ = 2176782336

and the probability of breaking it will be 0.004%

were in all cases you must think that each attempt made by the spyware is a different combination, so the one million tries are different combinations.