Question

probabilities with passwords
a 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.
what is the approximate probability that a password will have no vowels?
what is the approximate probability that the first letter of the password will be m?

Explanation:

Part 1: Probability of no vowels

Step 1: Determine total letters and vowels

There are 26 letters in the alphabet. Vowels are a, e, i, o, u (5 vowels), so consonants are \(26 - 5=21\). The password has 5 non - repeating letters. The total number of ways to choose 5 non - repeating letters from 26 is given by the permutation formula \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 26\) and \(r=5\). So total number of possible passwords \(N = P(26,5)=\frac{26!}{(26 - 5)!}=\frac{26!}{21!}=26\times25\times24\times23\times22\).

The number of ways to choose 5 non - repeating consonants (no vowels) is \(M = P(21,5)=\frac{21!}{(21 - 5)!}=\frac{21!}{16!}=21\times20\times19\times18\times17\).

Step 2: Calculate the probability

The probability \(P(\text{no vowels})=\frac{M}{N}=\frac{21\times20\times19\times18\times17}{26\times25\times24\times23\times22}\)

\[

$$\begin{align*} \frac{21\times20\times19\times18\times17}{26\times25\times24\times23\times22}&=\frac{21}{26}\times\frac{20}{25}\times\frac{19}{24}\times\frac{18}{23}\times\frac{17}{22}\\ &=\frac{21}{26}\times\frac{4}{5}\times\frac{19}{24}\times\frac{18}{23}\times\frac{17}{22}\\ &\approx0.8077\times0.8\times0.7917\times0.7826\times0.7727\\ &\approx0.28 \end{align*}$$

\]

Part 2: Probability the first letter is 'm'

Step 1: Analyze the first - letter choice

For the first letter of the password, there are 26 possible letters (since we are choosing the first letter of a 5 - letter non - repeating password, but for the first letter, we just consider all 26 letters). We want the first letter to be'm', so there is 1 favorable outcome (choosing'm') out of 26 possible outcomes.

Step 2: Calculate the probability

The probability \(P(\text{first letter is }m)=\frac{1}{26}\approx0.038\)

Answer:

s:

• Probability of no vowels: \(\approx0.28\)
• Probability first letter is \(m\): \(\approx0.038\) (or \(\frac{1}{26}\))