Chapter 13.8, Problem 3GP

To determine

To calculate: To find the probability of randomly choosing a $5$ letter password

Answer to Problem 3GP

Probability of randomly choosing a $5$ letter password that consists of only vowels is $0.03\%$

Explanation of Solution

Given information: $5$ letter password consists of only vowels if each vowel can be used more than once

Formula Used:

Probability is defined as the likelihood of an event.

Probability is calculated as

  $\mathrm{Probability}=\frac{Event}{Outcomes}$

Calculation:

Given that $5$ letter password consists of only vowels if each vowel can be used more than once

Experiment is defined as choosing a $5$ letter password

Sample space is calculated as

  $\begin{array}{l}S=26\times 26\times 26\times 26\times 26\\ S=11881376\end{array}$

Password consist of only vowels and each vowel can be used more than once. Total number of such possible passwords is

  $\begin{array}{l}N=5\times 5\times 5\times 5\times 5\\ N=3125\end{array}$

Also, given that out of $20$ people, $12$ said yes.

Now, probability of choosing a $5$ letter password that consists of only vowels if each vowel can be used more than once is calculated as:

  $\begin{array}{l}P=\frac{3125}{11881376}\\ P=0.026\%\\ P=0.03\%\end{array}$

Conclusion:

Hence, required probability is $0.03\%$