Question

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 probability that the units digit is 7 is 0.1. The probability that the hundreds digit is 7 is \( \approx 0.1111 \).

Step-by-step solution

Understanding the Problem

We need to find the probability that a randomly chosen three-digit number has a units digit of 7 and also the probability that it has a hundreds digit of 7.

Total Number of Three-Digit Numbers

First, calculate the total number of three-digit numbers. Three-digit numbers range from 100 to 999. Therefore, the total number of three-digit numbers is: \[ 999 - 100 + 1 = 900 \]

Counting Favorable Outcomes for Units Digit

Now, we need to count the number of three-digit numbers where the units digit is 7. Three-digit numbers range from 100 to 999, and the units digit can be 7 for each hundred (107, 117, ..., 997). Each hundred has exactly one number ending in 7, and there are 900 total numbers. Thus, there are 90 favorable outcomes.

Calculating Probability for Units Digit

The probability that a randomly chosen three-digit number has the units digit as 7 is: \[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{90}{900} = \frac{1}{10} = 0.1 \]

Counting Favorable Outcomes for Hundreds Digit

Now, count the number of three-digit numbers where the hundreds digit is 7. The hundreds digit can be 7 for the range 700 to 799. Therefore, there are exactly 100 numbers in this range.

Calculating Probability for Hundreds Digit

The probability that a randomly chosen three-digit number has the hundreds digit as 7 is: \[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{100}{900} = \frac{1}{9} \approx 0.1111 \]

Key concepts

Understanding Three-Digit Numbers

A three-digit number is any number between 100 and 999. This range is inclusive, meaning it includes both 100 and 999. Let's break down why this range results in exactly 900 numbers. You start with 100 and go up to 999. To find the total count, you calculate: \[ 999 - 100 + 1 = 900 \] This calculation shows us that there are exactly 900 three-digit numbers. Knowing the total count of possible outcomes is essential for calculating probabilities.

Favorable Outcomes

Favorable outcomes are the specific outcomes we're interested in when dealing with probability. In this problem, we have two different scenarios to consider. First, let's look at the units digit of 7. Any three-digit number can end in a 7, like 107, 117, 127, and so on. In each set of hundreds, there's exactly one number that ends in a 7. Since there are 900 three-digit numbers, and each hundred block contains exactly one number ending in 7, we have: \[ \text{90 favorable outcomes for units digit as 7} \] This number comes from 900 total numbers divided by 10 (since there are 10 digits in each hundred block).

Now, for the hundreds digit being 7, we consider the range from 700 to 799. Every number in this range has a hundreds digit of 7. So, there are exactly 100 favorable outcomes for having the hundreds digit as 7.

Step-by-Step Solution

Let's put everything together.

• First, calculate the total number of three-digit numbers, which is 900.


• For the units digit being 7, we found there are 90 favorable outcomes. So, the probability is: \[ \text{Probability} = \frac{90}{900} = \frac{1}{10} = 0.1 \]


• For the hundreds digit being 7, we found there are 100 favorable outcomes. The probability here is: \[ \text{Probability} = \frac{100}{900} = \frac{1}{9} \approx 0.1111 \]

By breaking down each step clearly, you can see how each part of the problem is approached and solved. This systematic approach helps ensure you don't miss any critical details when computing probabilities.