ZIP Code If you randomly select five digits, each between 0 and 9, with repetition allowed, what is the probability you will get the author’s ZIP code?

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with calculating the probability of randomly selecting a specific ZIP code, which consists of five digits. Each digit can range from 0 to 9, and repetition is allowed.
2. Step 2: Determine the total number of possible outcomes. Since there are 10 possible digits (0 through 9) for each of the five positions in the ZIP code, the total number of combinations is calculated as \( 10^5 \). This represents all possible five-digit combinations.
3. Step 3: Identify the number of favorable outcomes. There is only one specific ZIP code that matches the author's ZIP code, so the number of favorable outcomes is 1.
4. Step 4: Calculate the probability. The probability of selecting the author's ZIP code is the ratio of favorable outcomes to total outcomes, expressed as \( P = \frac{1}{10^5} \).
5. Step 5: Simplify the probability expression if needed. The probability can be left in its fractional form or converted to a decimal for interpretation, but the calculation stops here without evaluating the final numerical result.

Key Concepts

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it quantifies the chance of randomly selecting a specific five-digit ZIP code from all possible combinations of digits. Understanding probability is essential for calculating the chances of obtaining the author's ZIP code when selecting digits randomly.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. In this scenario, it helps determine the total number of possible five-digit combinations that can be formed using digits 0 through 9, with repetition allowed. This total is crucial for calculating the probability of selecting the specific ZIP code.

Independent Events

Independent events are outcomes where the occurrence of one event does not affect the occurrence of another. In the case of selecting digits for a ZIP code, each digit is chosen independently, meaning the choice of one digit does not influence the others. Recognizing this independence is vital for accurately calculating the overall probability of forming the author's ZIP code.