Question

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? 0.30935 what is the approximate probability that the first letter of the password will be m? 0.00160 0.03846 0.19231 0.80769

Explanation:

First Sub - Question (Probability of no vowels)

Step 1: Determine total number of possible passwords

The password has 5 non - repeating letters. The total number of ways to choose 5 non - repeating letters from 26 letters is given by the permutation formula \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 26\) and \(r=5\). So \(P(26,5)=\frac{26!}{(26 - 5)!}=\frac{26!}{21!}=26\times25\times24\times23\times22\).

Step 2: Determine number of passwords with no vowels

Vowels in English are \(a,e,i,o,u\) (5 vowels). So the number of non - vowel letters is \(26- 5=21\). The number of ways to choose 5 non - repeating non - vowel letters is \(P(21,5)=\frac{21!}{(21 - 5)!}=\frac{21!}{16!}=21\times20\times19\times18\times17\).

Step 3: Calculate the probability

The probability \(P(\text{no vowels})=\frac{P(21,5)}{P(26,5)}=\frac{21\times20\times19\times18\times17}{26\times25\times24\times23\times22}\)
\[

$$\begin{align*} \frac{21\times20\times19\times18\times17}{26\times25\times24\times23\times22}&=\frac{21\times20\times19\times18\times17}{26\times25\times24\times23\times22}\\ &=\frac{21}{26}\times\frac{20}{25}\times\frac{19}{24}\times\frac{18}{23}\times\frac{17}{22}\\ &\approx0.8077\times0.8\times0.7917\times0.7826\times0.7727\\ &\approx0.3094 \end{align*}$$

\]

Second Sub - Question (Probability first letter is \(m\))

Step 1: Total number of choices for first letter

There are 26 possible letters for the first position (since we start with choosing the first letter, and later letters are chosen without repetition, but for the first letter, we have 26 options).

Step 2: Number of favorable choices for first letter

There is only 1 favorable choice (the letter \(m\)).

Step 3: Calculate the probability

The probability \(P(\text{first letter is }m)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{1}{26}\approx0.0385\) (which is approximately \(0.03846\))

Answer:

s:

• Probability of no vowels: \(\approx0.3094\)
• Probability first letter is \(m\): \(0.03846\) (corresponding to the option with this value)