Chapter 4.3, Problem 89STP

To determine

To explain: The probability that password begins with a consonant.

  $\begin{array}{l}\text{F}\text{. }\frac{21}{26}\text{ H}\text{. }\frac{21}{36}\\ \text{G}\text{. }\frac{21}{35}\text{ J}\text{. }\frac{5}{36}\end{array}$

Answer to Problem 89STP

The probability is $\frac{21}{36}$ . Option H. is correct.

Explanation of Solution

Given information:

A 5-character password can contain the number 0 through 9 and $26$ letters of the alphabet. None of the letters is repeated.

Formula used:

Probability of an event is calculated as number of favorable outcomes divided by total number of outcomes which is mathematically expressed as,

  $P\left(E\right)=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$

Calculation:

Consider the given information that “A 5-character password can contain the number 0 through 9 and $26$ letters of the alphabet. None of the letters is repeated.”

Recall that probability of an event is calculated as number of favorable outcomes divided by total number of outcomes which is mathematically expressed as,

  $P\left(E\right)=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$

Since, password can take $10$ digits and $26$ letters, the total number of outcomes is,

  $10+26=36$

Since, the password has to begin with a consonant, number of consonants in the alphabets is $21$ out of $26$ .

So, favorable outcomes are $21$ .

Probability that password begins with consonant is,

  $\begin{array}{c}P\left(E\right)=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}\\ P\left(E\right)=\frac{21}{36}\end{array}$

Thus, the probability is $\frac{21}{36}$ . Option H. is correct.