Problem 154 - Applied Statistics and Probability for Engineers

Question

A credit card contains 16 digits. It also contains the month and year of expiration. Suppose there are 1 million credit card holders with unique card numbers. A hacker randomly selects a 16-digit credit card number. (a) What is the probability that it belongs to a user? (b) Suppose a hacker has a \(25 \%\) chance of correctly guessing the year your card expires and randomly selects 1 of the 12 months. What is the probability that the hacker correctly selects the month and year of expiration?

Step-by-Step Solution

Answer

(a) Probability: \(10^{-10}\). (b) Probability: 0.02083.

Step 1: Understanding the Card Number Probability

We have 1 million unique credit card numbers assigned to different users. This means there are 1,000,000 possibilities for a card number to belong to a user. The total number of possible 16-digit numbers is between \(10^{15}\) and \(10^{16}\), depending on the first digit (which cannot be 0). Most often, when discussing credit cards, we assume a full range of \(10^{16}\) due to the prefix constraints for different card types, but for simplicity, let’s assume exactly \(10^{16}\) possibilities are used here. The probability that a randomly selected 16-digit card number matches one of these specific 1 million user numbers is calculated by dividing the number of successful outcomes (1 million) by the number of possible outcomes (\(10^{16}\)).

Step 2: Calculating the Probability for Part (a)

The number of valid credit card numbers that belong to users is 1,000,000 and the total number of possible 16-digit numbers is \(10^{16}\). Therefore, the probability that a randomly selected 16-digit number matches one of these is:\[ P(\text{Correct Card}) = \frac{1,000,000}{10^{16}} = 10^{-10}\]

Step 3: Understanding the Month and Year Expiration Probability

The problem states that the hacker has a \(25\%\) chance of correctly guessing the year your card expires. This equals a probability of \(0.25\). The hacker also needs to correctly guess 1 out of 12 months. The probability of selecting the correct month is therefore \(\frac{1}{12} \approx 0.0833\).

Step 4: Calculating the Probability for Part (b)

To find the probability that the hacker correctly guesses both the month and the year, multiply the probability of guessing the year correctly by the probability of guessing the month correctly:\[ P(\text{Correct Month and Year}) = 0.25 \times \frac{1}{12} = 0.25 \times 0.0833 = 0.02083 \]

Key Concepts

Credit Card Security

Credit card security is crucial in protecting the sensitive financial information of individuals. Credit cards are issued with unique 16-digit numbers.
These numbers are not random but are designed with specific prefixes that denote the issuing bank and type of card. These prefixes, combined with additional identifying numbers, make credit card numbers hard to guess or reproduce.

• Each card has a unique number, making unauthorized attempts to use an unknown number highly unsuccessful.
• The expiration date, often a combination of month and year, further adds a layer of complexity and security.

The key to credit card security is ensuring that your card details remain confidential and inaccessible to unauthorized entities.

Random Selection

Random selection is a fundamental concept in probability where each possible outcome has an equal chance of being chosen.
In the context of credit cards, a hacker making a guess is engaging in a random process. With a large pool of possibilities, the likelihood of randomly choosing the correct number is extremely low.
For instance, consider a hacker randomly guessing a 16-digit credit card number. Since the first digit cannot be 0, and assuming a full range of numbers, there are approximately \(10^{16}\) potential combinations.

• This means the choice is made from an enormous set, making the probability of a correct guess minuscule.
• Random processes ensure that, without insider knowledge or advanced predictions, the odds remain very unfavorable for unauthorized guesses.

Event Probability

Event probability determines how likely an occurrence is within a set of possible events. It's calculated by dividing the number of successful outcomes by the total number of possible outcomes.
For example, if a hacker selects a 16-digit number at random from all possible combinations, the probability of correctly guessing a valid user's card number is calculated as shown:

• There are 1 million valid card numbers and about \(10^{16}\) possible 16-digit combinations.
• The probability the hacker guesses right is \(\frac{1,000,000}{10^{16}} = 10^{-10}\).

This low probability highlights the effectiveness of such a large range in maintaining security against random guesses.

Probability Calculation

Probability calculation involves determining the chance of multiple independent events occurring together.
This often involves multiplying the probabilities of individual events. In the credit card scenario, a hacker trying to guess both the month and year of card expiration needs to calculate each separately:

• The chance of guessing the year correctly, given as a 25% chance, is \(0.25\).
• The chance of selecting the correct month out of 12 is \(\frac{1}{12} \approx 0.0833\).
• Therefore, the joint probability of correctly guessing both the year and month is \(0.25 \times 0.0833 = 0.02083\).

This combined probability underscores the difficulty in correctly guessing by sheer chance, further safeguarding security through the structure and complexities built into the system.