Problem 66 - Applied Statistics and Probability for Engineers

Question

A credit card contains 16 digits between 0 and \(9 .\) However, only 100 million numbers are valid. If a number is entered randomly, what is the probability that it is a valid number?

Step-by-Step Solution

Answer

The probability is \(10^{-8}\) or \(0.00000001\).

Step 1: Calculate Total Possible Numbers

Find the total number of possible 16-digit credit card numbers. Each digit can be anything from 0 to 9, so there are 10 choices for each digit. Therefore, the total number of possible credit card numbers is given by \(10^{16}\).

Step 2: Identify Valid Numbers

We know from the problem statement that there are only 100 million valid credit card numbers. In numerical form, this is \(100,000,000\).

Step 3: Calculate Probability

To find the probability that a randomly chosen credit card number is valid, divide the number of valid numbers by the total number of possible numbers. Thus, the probability \(P\) is \(P = \frac{100,000,000}{10^{16}}\).

Step 4: Simplify the Probability

Calculate the probability by performing the division. Simplifying \(\frac{100,000,000}{10^{16}}\), we find \(P = 10^{-8}\). Alternatively, this is \(0.00000001\) as a decimal.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting and arranging objects. In problems involving large numbers of possibilities, such as credit card numbers, combinatorial methods are essential. When dealing with credit card numbers, each digit can be one of several choices, specifically between 0 and 9.

This makes it suitable to employ the rule of product, a fundamental combinatorial principle. It states that if there are \(n\) ways to perform one action and \(m\) ways to perform another, there are \(n \times m\) ways to perform both actions. This principle helps us calculate total possibilities for multiple sequences or options simultaneously.

• For credit cards: each of the 16 digits can be any number from 0 to 9.
• This results in \(10^{16}\) total possible combinations.

The understanding of combinatorics simplifies the process of evaluating such massive numbers of possibilities.

Credit Card Number

A credit card number is a sequence of 16 digits, used primarily as a unique identifier for processing payments. Each digit can range from 0 to 9, and the sequence must adhere to certain validity checks.

The structure ensures security and uniqueness of each card. This complexity makes the chance of randomly guessing a valid number extremely low, which is crucial for the security against fraudulent activities. Understanding the formation:

• Each of the 16 positions has 10 possible numbers (0 through 9).
• The total possible combinations of digits amount to \(10^{16}\).

However, not every combination is used. The verification process filters out invalid numbers, reducing the actual count significantly.

Probability Calculation

Probability helps us quantify the chance of an event occurring. When dealing with large numbers, like credit cards, it's vital to accurately calculate this probability.

To determine the likelihood that a randomly selected credit card number is valid, we divide the number of valid numbers by the total number of possible combinations:

• Total possible numbers: \(10^{16}\).
• Valid numbers: 100 million or \(100,000,000\).
• Probability equation: \[P = \frac{100,000,000}{10^{16}}\]

Simplifying this fraction results in a probability of \(10^{-8}\) or \(0.00000001\). This minuscule probability accentuates how unlikely it is to guess a valid number by random chance.

Valid Numbers

Valid credit card numbers are uniquely assigned sequences that pass specific validation algorithms. Not all 16-digit combinations form valid numbers; many are eliminated through these checks.

The actual criteria for valid numbers often include algorithms like the Luhn check, which ensure the integrity and legitimacy of the number. For simplicity:

• Although there are \(10^{16}\) combos, only 100 million are valid.
• This reduces the count significantly from all possible numbers generated by simple combinatorial logic.

Understanding which numbers are deemed valid ensures reliable usage and imparts a sense of security necessary for such financial tools.