Question

Motorola has developed an edible password pill. When the pill is ingested, it is activated by the acid in your stomach emitting an 18-bit signal which is strong enough to communicate and unlock all your devices. The edible password pill won’t be available in the near future, but it is already FDA approved. A computer system uses passwords that contain exactly six characters, and each character is one of 26 lower case letters (a, z), 26 upper case letters (A, Z), and 10 integers (0, 9). Let S denote the set of all possible passwords, A, denote the event that the password contains only letters, and, B, be the event that the password contains only lower case letters. Suppose that all passwords in S are equally likely. Determine the probability for each of the following:

(a) The password contains all lower case letters given that it contains only letters.
(b) The password contains at least one upper case letter given that it contains only letters.

(c) The password contains only even numbers given that is contains all numbers.

(d) The password contains exactly three or four numbers.

Answer 1

We can use 26 lower case letters, 26 upper case letters and 0-9 digits.

So, total ways one place can be fill = 26+26+10 = 62

Password length = 6 characters

Total possible password = 62*62*62*62*62*62 = 62^6

A) P(only lower cases | only letters) =

26 because only lower cases so total ways for one place = 26

52 because only letters so total ways for one place = 26 + 26 = 52

B) P(At least one upper case | only letters) = 1 - P(only lower cases | only letter) =

C) P(only even numbers | only numbers) =

5 because only even numbers so total ways for one place = 5 {0, 2, 4, 6, 8}

10 because only numbers so total ways for one place = 10

D) P(exactly 3 or 4 numbers) = P(3 numbers) + P(4 numbers)

= P(3 numbers, 3 letters) + P(4 numbers, 2 letters)

=

62^6 is the total passwords

6C3 because select 3 places out of 6 in which it will take numbers and same way 6C4.

=  (approx).