A computer system uses passwords that contain exactly seven characters, and each

Question

A computer system uses passwords that contain exactly seven characters, and each character is 1 of the 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Let denote the set of all possible passwords, and let A and B denote the events that consist of passwords with only letters or only integers, respectively.Determine the probability for each of the following: (e) Password contains at least 1 lowercase letter given that it contains only letters (f) Password contains only odd numbers given that it contains all numbers

Explanation / Answer

(e) If the password contains only letters there are 52 possibilities for each character of password, including uppercase and lowercase letters. If it contains only lowercase letters each character will have 26 possibilities.

Hence given that password contains only letters, the probability that each character will contain lowercase letters is = 26 / 52 = 0.5.

So the required conditional probability that password contains at least one lowercase letters given that it contains only letters = 0.5 + (0.5) ^2 +....+ (0.5)^7. (Ans).

(f) Probability that the password contains odd number in each character given that it contains only numbers = 5 / 10 = 0.5.

Hence the required conditional probability = (0.5)^ 7 = 0.0078. (Ans).

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