Problem 67 - Applied Statistics and Probability for Engineers

Question

Suppose your vehicle is licensed in a state that issues license plates that consist of three digits (between 0 and 9) followed by three letters (between \(A\) and \(Z\) ). If a license number is selected randomly, what is the probability that yours is the one selected?

Step-by-Step Solution

Answer

The probability is \(\frac{1}{17,576,000}\).

Step 1: Calculate Total Number of License Plates

First, calculate the total number of license plates. There are 10 possible digits (0-9) for each of the three digit positions, so for the three digits, there are \[10 \times 10 \times 10 = 10^3\] combinations. There are 26 possible letters (A-Z) for each of the three letter positions, so for the three letters, there are \[26 \times 26 \times 26 = 26^3\]combinations. Thus, the total number of license plates is\[10^3 \times 26^3.\]

Step 2: Calculate Probability of Your License Plate Being Selected

The probability that one specific license plate is selected is the reciprocal of the total number of license plates possible. With\[10^3 \times 26^3\]possible license plates, the probability is given by \[\frac{1}{10^3 \times 26^3}.\]

Step 3: Simplify the Calculation

Compute the value of the above expression. First calculate \[10^3 = 1000\] and \[26^3 = 17576,\] so that\[10^3 \times 26^3 = 1000 \times 17576 = 17576000.\] Thus, the probability is \[\frac{1}{17576000}.\]

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, combinations, and permutations. In this exercise, we apply combinatorics to count the possible outcomes for creating a license plate. Each license plate comprises three digits and three letters. For the digits, we have ten choices (0 to 9) and for the letters, we have twenty-six choices (A to Z). Therefore, for each of the three digit places, we calculate the total as follows:

• 10 choices for the first position
• 10 choices for the second position
• 10 choices for the third position

Similarly, for the letters, we compute:

• 26 choices for the first position
• 26 choices for the second position
• 26 choices for the third position

Thus, the total number of possible combinations for the license plate is calculated by multiplying these possibilities: \[10^3 \times 26^3\]. This process helps determine the entire set of outcomes, a fundamental skill in combinatorics. Understanding this helps solve many real-world problems related to counting and arrangement.

Random Selection

Random selection plays a key role in determining probabilities. When something is chosen randomly, each option has an equal chance of being selected. In our case, we're determining the probability of a specific license plate being the one chosen out of millions.
Imagine reaching into a huge bowl of marbles, where each marble represents a different license plate. Each marble has an equal chance of being picked. The randomness ensures fairness and unpredictability in the selection process.
Random selection is an essential concept in probability and statistics. It is commonly used in sampling methods, where a representative sample is chosen from a larger population. By applying the concept of randomness, we make sure that every outcome is equally likely, and avoid any bias during selection.

License Plate Probability

The probability of any specific license plate being selected in our exercise involves understanding the ratio of one desired outcome to the total possible outcomes. This is calculated by taking the reciprocal of the total number of possible license plates.
We first determine the total number of possible outcomes, which are all the unique license plates that can be formed. Using our combinatorics knowledge, we already calculated this to be \(10^3 \times 26^3 = 17576000\). Hence, the probability of selecting any single license plate is \[\frac{1}{17576000}.\]
This fraction represents the incredibly small chance of any specific plate being chosen at random. Understanding this helps appreciate how probabilities quantify uncertainty and express how likely events are to occur. Probability is widely applicable, offering insights into fields ranging from statistics to natural sciences.