A password has to use the following format: LDDDDLLL, where L can be any of the upper case letters in the alphabet, and D can be any digit including 0. None of the digits or letters can repeat. What is the probability that a random password uses only vowels (A,E,I,O,U) and odd numbers (1,3,5,7,9)? Show work but do not evaluate. What is the probability that the password spells the word “MATH?”

We have four letters, so the probability that one letter is a vowel is 5/26 (we have 5 vowels in a total of 26 letters), then the second letter has a probability of 4/25 of being a vowel (1 vowel used), and so on (third letter being vowel = 3/24 and fourth letter being vowel = 2/23)

Then, for the digits, we do the same, one digits has 5/10 probability of being odd, then the second digit has 4/9, the third has 3/8 and the fourth has 2/7.

So the final probability would be:

[tex]P = \frac{5}{26} *\frac{4}{25}* \frac{3}{24}* \frac{2}{23} *\frac{5}{10}* \frac{4}{9} *\frac{3}{8} *\frac{2}{7}[/tex]

To find the probability that the password spells the word “MATH", each letter has to be a specific letter, so the first letter has 1/26 probability, the second has 1/25, and so on:

[tex]P = \frac{1}{26} *\frac{1}{25}* \frac{1}{24}* \frac{1}{23}[/tex]

MrRoyal MrRoyal · Answer 2 · 2022-03-29T16:11:33+02:00

The number of passwords with only vowels and odd numbers is 1/125580 and the probability that a password spells MATH is 1/358800

How to determine the probability

The password format is given as:

LDDDDLLL

Since no character can be repeated, then the total number of different password using the format is:

Total = 26 * 10 * 9 * 8 * 7 * 25 * 24 * 23

Total = 1808352000

Also using the format, the number of passwords with only vowels and odd numbers is:

Total = 5 * 5 * 4 * 3 * 2 * 4 * 3 * 2

Total = 14400

The probability is then calculated as:

[tex]p = \frac{14400}{1808352000}[/tex]

Simplify

[tex]p = \frac{1}{125580}[/tex]

The number of password that spells MATH is:

Total = 1 * 10 * 9 * 8 * 7 * 1 * 1 * 1

Total = 5040

So, the probability that a password spells MATH is:

[tex]p = \frac{5040}{1808352000}[/tex]

Simplify

[tex]p = \frac{1}{358800}[/tex]

Hence, the probability that a password spells MATH is 1/358800