Question

Set up an appropriate sample space for each of Problems 1.1 to 1.10 and use itto solve the problem. Use either a uniform or non-uniform sample space or try both.

If you select a three-digit number at random, what is the probability that the units digit is 7? What is the probability that the hundreds digit is 7?

Short answer

The required sample space is $100 , 101 , \dots , 999$

The probability that the units digit is 7 is$\frac{1}{10}$ andthe probability that the hundreds digit is 7 is$\frac{1}{9}$.

Step-by-step solution

Definition of Probability

The probability of any event is defined as the ratio of the number of outcomes associated with the event to the total number of possible outcomes. The probability of a particular event is always less than or equal to 1.

Creation of the sample space

A three-digit number lies from and thus, there are 900 three-digit numbers. So, the sample space for the given problem is all the three-digit number from 100 to 999, that is expressed as follows,

$100 , 101 , \dots , 999$

Each digit of the obtained sample space has an equal probability of $\frac{1}{900}$.

Determination of the probability that the units digit is 7

A three-digit number which end with 7 are 90 namely$\left(107,117,\cdots ,197,207,\cdots ,297,\cdots ,907,917,\cdots ,997\right)$

with each having a probability of$\frac{1}{900}$ .

Find the probability that the units digit is 7 by adding the probabilities of each possible outcomes, that is 90 times $\frac{1}{900}$.

$\begin{array}{rcl}p& =& 90\times \frac{1}{900}\\ & =& \frac{1}{10}\\ & & \end{array}$

Thus, the probability that the units digit is 7is$\frac{1}{10}$ .

Determination of the probability that the hundreds digit is 7

A three-digit number which starts with 7 are 100that are$\left(700,701,\cdots ,799\right)$ with each having a probability of $\frac{1}{900}$.

Find the probability that the hundreds digit is 7by adding the probabilities of each possible outcomes, that is 100 times$\frac{1}{900}$ .

role="math" localid="1655788370190" $\begin{array}{rcl}p& =& 100\times \frac{1}{900}\\ & =& \frac{1}{9}\\ & & \end{array}$

Thus, the probability that the hundreds digit is 7 is $\frac{1}{9}$.