Question

In Exercises 5–36, express all probabilities as fractions.

Identity Theft with Credit Cards Credit card numbers typically have 16 digits, but not all of them are random.

a. What is the probability of randomly generating 16 digits and getting your MasterCard number?

b. Receipts often show the last four digits of a credit card number. If only those last four digits are known, what is the probability of randomly generating the other digits of your MasterCard number?

c. Discover cards begin with the digits 6011. If you know that the first four digits are 6011 and you also know the last four digits of a Discover card, what is the probability of randomly generating the other digits and getting all of them correct? Is this something to worry about?

Short answer

a. The probability of getting the correct MasterCard number is$\frac{1}{{10}^{16}}$.

b. The probability of getting the correct MasterCard number if the last four digits are known is$\frac{1}{{10}^{12}}$.

c. The probability of getting the correct MasterCard number if the first and the last four digits are known is $\frac{1}{{10}^8}$.

No, there is nothing to worry about the credit card number being randomly guessed as the probability is extremely low.

Step-by-step solution

Given information

A credit card number consists of 16 digits.

State the counting rule

When an event occurs in multiple steps, thenumber of unique ways in which it can occur is the product of the number of ways in which each step of the event occurs.

Compute the probability to obtain a specific MasterCard number

a.

The total number of digits from 0 to 9 is 10.

The number of possible ways to write a 16-digit number is

$10\times 10\times 10\times ......\times 10\left(16\text{times}\right)={\left(10\right)}^{16}$

The number of ways of writing the correct MasterCard number is 1.

The probability of getting the correct MasterCard number is

$P\left(\text{correct}\text{number}\right)=\frac{1}{{10}^{16}}$

Therefore, the probability of getting the correct MasterCard number is $\frac{1}{{10}^{16}}$.

Compute the probability when only the last four digits are known

b.

The last four digits of the number are known, and the rest 12 digits are to be selected.

The number of ways 12 digits of card number can be selected is one.

The number of ways in which any one of the remaining 12 digits can be chosen is 10.

The total number of ways for forming the remaining 12 digits of MasterCard is

$$\begin{gathered} 10\times 10\times .......\times 10\times 1\times 1\times 1\times 1\left(16\text{terms}\right)={10}^{12}\times 1 \\ ={10}^{12} \end{gathered}$$

The probability of getting the correct MasterCard number if the last four digits are known is

$P\left(\text{correct}\text{number}\right)=\frac{1}{{10}^{12}}$

Therefore, the probability of getting the correct MasterCard number when the last four digits are known is $\frac{1}{{10}^{12}}$.

Compute the probability when the first and last four digits are known

c.

The first and the last four digits of the number are known.

So, the number of ways in which a unique card number can be selected is 1.

The number of ways in which any one of the remaining eight digits can be chosen is 10.

The total number of ways is

$$\begin{gathered} 1\times 1\times 1\times 1\times 10\times 10\times .......\times 10\times 1\times 1\times 1\times 1\left(16\text{terms}\right)=1\times {10}^8\times 1 \\ ={10}^8 \end{gathered}$$

The probability of getting the correct MasterCard number if the first and the last four digits are known is

$P\left(\text{correct}\text{number}\right)=\frac{1}{{10}^8}$

Therefore, the probability of getting the correct MasterCard number when the first and the last four digits are known is $\frac{1}{{10}^8}$.

Compute the probability when only the last four digits are known

The probability of guessing the card number even when a certain set of digits is known is close to zero.

As the probabilities are extremely low (approximately equal to 0), it is almost impossible that the number of the MasterCard is randomly guessed by any stranger.

Thus, there is nothing to worry about.