Chapter 6.8, Problem 19E

i.

To determine

To find the number of possible outcomes.

Answer to Problem 19E

There are $100000$ different numbers which can begin with $79$ .

Explanation of Solution

Given:

There is a $7$ digit phone number.

The phone number begins with the number $79$ .

Formula Used:

If one event can occur in $m$ ways, and for each of these ways a second event can occur in $n$ ways, then the number of ways that the two events can occur together is $m\times n$ .

The multiplication principle can be extended to three or more events.

Calculation:

In each of the $5$ other digits a total of any $10$ numbers ranging from $0$ to $9$ can be placed.

Therefore,

Total number of possible outcomes for the other $5$ digit is:

  $\begin{array}{l}=10\times 10\times 10\times 10\times 10\\ =100000\end{array}$

Hence,

There are $100000$ different numbers which can begin with $79$ .

ii.

To determine

To find the number of possible outcomes.

Answer to Problem 19E

There are $10000$ different numbers which can begin with $798$ .

Explanation of Solution

Given:

There is a $7$ digit phone number.

The phone number begins with the number $798$ .

Formula used:

If one event can occur in $m$ ways, and for each of these ways a second event can occur in $n$ ways, then the number of ways that the two events can occur together is $m\times n$ .

The multiplication principle can be extended to three or more events.

Calculation:

In each of the $4$ other digits a total of any $10$ numbers ranging from $0$ to $9$ can be placed.

Therefore,

Total number of possible outcomes for the other $4$ digit is:

  $\begin{array}{l}=10\times 10\times 10\times 10\\ =10000\end{array}$

Hence,

There are $10000$ different numbers which can begin with $798$ .

iii.

To determine

To find the number of correct digit.

Answer to Problem 19E

  $5$ digit.

Explanation of Solution

Given:

There is a $7$ digit phone number.

Formula used:

  $P\left(\text{event}\right)=\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}$

Calculation:

In each of the $7$ digits of a phone number a total of any $10$ numbers ranging from $0$ to $9$ can be placed.

Total number of possible outcomes for the $7$ digits phone number is:

  $\begin{array}{l}=10\times 10\times 10\times 10\times 10\times 10\times 10\\ =10000000\end{array}$

Probability of guessing a number correctly $=\frac{1}{100}$

Therefore,

Total number of phone digit should be known to get probability $\frac{1}{100}$ is:

  $\begin{array}{l}P\left(\text{event}\right)=\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}\\ \frac{1}{100}=\frac{\text{Number of favorable outcomes}}{10000000}\\ 100000=\text{ Number of favorable outcomes}\end{array}$

The total outcomes one should know to get the probability $\frac{1}{100}$ is 10000, and as it is known that each digit of a phone number range from $0$ to $9$ which is consists of $10$ numbers. Therefore one should know $10\times 10\times 10\times 10\times 10$ that it requires $5$ correct digits of $7$ digit of phone numbers to get the probability of $\frac{1}{100}$ .

Hence,

There are $5$ digit of $7$ digit phone number one should know to randomly guess the number correctly and the probability comes $\frac{1}{100}$ .