Problem 72 - Algebra and Trigonometry

Question

Some three-digit numbers, such as 101 and \(313,\) read the same forward and backward. If you select a number from all threedigit numbers, find the probability that it will read the same forward and backward.

Step-by-Step Solution

Answer

The probability of a randomly chosen three-digit number reading the same forward and backward (being a palindrome) is 0.1 or 10%.

Step 1: Finding Total Three-Digit Numbers

There are 900 total three-digit numbers, ranging from 100 to 999.

Step 2: Finding Total Palindrome Numbers

In a palindromic three-digit number, the first (hundreds place) and the last digit (ones place) must be the same. This allows for 9 options (1-9, as 0 cannot be the first digit of a three-digit number) for the first number and the last digit simultaneously. The middle digit (tens place) can be any digit from 0 to 9, which gives 10 options. Therefore, the total palindrome three-digit numbers are \(9 \times 10 = 90\).

Step 3: Calculate Probability

Probability is calculated as 'favourable outcomes' divided by 'total outcomes'. The favourable outcomes here are the 90 palindrome numbers and the total outcomes are the 900 total three-digit numbers. So, the required probability is \(\frac{90}{900} = \frac{1}{10} = 0.1\).

Key Concepts

Three-Digit Numbers

Three-digit numbers are numbers that range from 100 to 999. Each number consists of three digits placed in the hundreds, tens, and ones positions. This range gives us exactly 900 unique numbers. Here’s why: the smallest three-digit number is 100, and the largest is 999.
To find out how many numbers there are in this range, we simply subtract the smallest number (100) from the largest number (999) and add one to include the number 100 in our count.

• The formula for this is: \[999 - 100 + 1 = 900\]
• So, there are 900 three-digit numbers altogether.

Understanding this concept is essential when we calculate probabilities involving three-digit numbers.

Palindrome Numbers

A palindromic number is a number that reads the same forward and backward, meaning it mirrors itself. For three-digit numbers, the digits in the hundreds and ones place must be identical, while the digit in the tens place can vary freely.
To form a three-digit palindrome:

• The hundreds and ones place must have the same digit. This digit can be any from 1 to 9, because a three-digit number cannot begin with 0. This gives us 9 possible choices.
• The tens place, meanwhile, can contain any of the 10 digits from 0 to 9.
• Therefore, the number of three-digit palindromic numbers can be calculated by multiplying the number of choices for the hundreds/ones place with the choices for the tens place: \[9 \times 10 = 90\]

Thus, there are 90 three-digit palindrome numbers.

Favourable Outcomes

In probability, an outcome is any possible result of an action. A favourable outcome is a result that fits the criteria we're interested in—in this case, a number reading the same forward as backward.
When calculating the probability that a randomly chosen three-digit number is a palindrome, we need to:

• Identify the favourable outcomes. For this problem, they are the 90 palindromic numbers, as these meet our desired condition.
• Determine the total possible outcomes, which are the total three-digit numbers, numbered from 100 to 999. We have already determined this as 900.

The formula for probability is: \[\text{Probability} = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}\]Applying this to our situation gives us: \[\frac{90}{900} = \frac{1}{10} = 0.1\]Thus, the probability that a randomly selected three-digit number is a palindrome is 0.1 or 10%.