Probability of a Three-Digit Number Containing the Digit 8 at Most Once

When selecting a three-digit number randomly, what is the probability that it contains the digit 8 at most once? This article will delve into the calculations behind this probability question, analyzing various cases and providing a detailed explanation.

Introduction

The problem requires us to determine the probability of a randomly selected three-digit number containing the digit 8 at most once. This involves understanding the range of three-digit numbers, breaking down the scenarios where the digit 8 appears, and calculating the probability accurately.

Total Number of Three-Digit Numbers

First, let's establish the total number of three-digit numbers. Three-digit numbers range from 100 to 999. Thus, the total count is:

999 - 100 1 900

Count of Three-Digit Numbers with the Digit 8 Appearing at Most Once

We need to consider two cases: numbers with no digit 8 and numbers with exactly one digit 8. We'll break this down step by step.

Case 1: No Digit 8

If there is no 8 in the number, the first digit (hundreds place) can be any digit from 1 to 9 except 8, giving us 8 options. The tens and units places can each be any digit from 0 to 9 except 8, giving us 9 options each. The total number of such numbers is:

8 * 9 * 9 648

Case 2: Exactly One Digit 8

For one digit 8, there are three positions: hundreds, tens, or units.

8 in the Hundreds Place

This scenario is not possible because it would make the number a two-digit number.

8 in the Tens Place

If 8 is in the tens place, the hundreds place can be any digit from 1 to 9 except 8, giving us 8 options. The units place can be any digit from 0 to 9 except 8, giving us 9 options. The total for this scenario is:

8 * 9 72

8 in the Units Place

If 8 is in the units place, the hundreds place can be any digit from 1 to 9 except 8, giving us 8 options. The tens place can be any digit from 0 to 9 except 8, giving us 9 options. The total for this scenario is:

8 * 9 72

Adding the scenarios where 8 appears once:

72 72 144

Total Numbers with at Most One 8

Now, we can combine both cases:

Numbers with no digit 8: 648 Numbers with exactly one digit 8: 144

The total number of three-digit numbers with the digit 8 appearing at most once is:

648 144 792

Calculating the Probability

The probability P that a randomly selected three-digit number contains the digit 8 at most once is:

P frac{792}{900}

Simplifying this fraction:

P 0.88

Conclusion

The probability that a randomly selected three-digit number contains the digit 8 at most once is 0.88 or 88%.