Question

A computer system uses passwords that contain exactly eight characters, and each character is one of the 26 lowercase letters (a–z) or 26 uppercase letters (a– z) or 10 integers (0–9). assume all passwords are equally likely. let a and b denote the events that consist of passwords with only letters or only integers, respectively. determine the following probabilities:

P( ∪ )

Answer 1

To find the probability of passwords made up of either only letters or only integers (P(A ∪ B)), we calculate the probability of each event separately and add them together, considering they are mutually exclusive.

Step-by-step explanation:

The question pertains to the calculation of probabilities for passwords consisting of different characters. To determine P(A ∪ B), we first need to calculate the probabilities of each event separately and then combine them, keeping in mind that A and B are mutually exclusive events (as a password cannot be made of only letters and only numbers simultaneously).

For event A (passwords with only letters): There are 26 lowercase plus 26 uppercase letters, giving us a total of 52 possible letter choices for each character. Thus, the probability of choosing a password with only letters is P(A) = (52/62)^8.

For event B (passwords with only integers): There are 10 possible integer choices for each character. Hence, the probability of choosing a password with only integers is P(B) = (10/62)^8.

Since A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B). After calculating the individual probabilities, we can add them together to get the total probability of choosing a password that consists of either only letters or only integers.