probabilities with passwords\na website randomly creates an initial password for people when they first sign…

probabilities with passwords\na website randomly creates an initial password for people when they first sign up for an account. the password consists of five letters, and cannot include numbers or special characters. the letters of the password cannot repeat.\nwhat is the approximate probability that a password will have no vowels?\nwhat is the approximate probability that the first letter of the password will be m?

probabilities with passwords\na website randomly creates an initial password for people when they first sign up for an account. the password consists of five letters, and cannot include numbers or special characters. the letters of the password cannot repeat.\nwhat is the approximate probability that a password will have no vowels?\nwhat is the approximate probability that the first letter of the password will be m?

Answer

Explanation:

Step1: Calculate total number of possible passwords

There are 26 letters in the alphabet. The number of ways to create a 5 - letter non - repeating password is given by the permutation formula (P(n,r)=\frac{n!}{(n - r)!}), where (n = 26) and (r=5). So (P(26,5)=\frac{26!}{(26 - 5)!}=\frac{26!}{21!}=26\times25\times24\times23\times22).

Step2: Calculate number of passwords with no vowels

There are 5 vowels ((a), (e), (i), (o), (u)), so there are (26-5 = 21) non - vowels. The number of 5 - letter non - repeating passwords with no vowels is (P(21,5)=\frac{21!}{(21 - 5)!}=\frac{21!}{16!}=21\times20\times19\times18\times17). The probability (P_1) that a password has no vowels is (P_1=\frac{P(21,5)}{P(26,5)}=\frac{21\times20\times19\times18\times17}{26\times25\times24\times23\times22}\approx0.29).

Step3: Calculate probability that first letter is (m)

If the first letter is fixed as (m), then we need to choose the remaining 4 letters from the remaining 25 letters. The number of ways to do this is (P(25,4)=\frac{25!}{(25 - 4)!}=\frac{25!}{21!}=25\times24\times23\times22). The probability (P_2) that the first letter is (m) is (P_2=\frac{P(25,4)}{P(26,5)}=\frac{25\times24\times23\times22}{26\times25\times24\times23\times22}=\frac{1}{26}\approx0.038).

Answer:

The approximate probability that a password will have no vowels is (0.29). The approximate probability that the first letter of the password will be (m) is (0.038).