Question

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?”

Answer 1

Answer:

Only vowels and odd numbers:

$P = \frac{5}{26} *\frac{4}{25}* \frac{3}{24}* \frac{2}{23} *\frac{5}{10}* \frac{4}{9} *\frac{3}{8} *\frac{2}{7}$

Spells math:

$P = \frac{1}{26} *\frac{1}{25}* \frac{1}{24}* \frac{1}{23}$

Step-by-step explanation:

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:

$P = \frac{5}{26} *\frac{4}{25}* \frac{3}{24}* \frac{2}{23} *\frac{5}{10}* \frac{4}{9} *\frac{3}{8} *\frac{2}{7}$

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:

$P = \frac{1}{26} *\frac{1}{25}* \frac{1}{24}* \frac{1}{23}$