91Ó°ÊÓ

A credit card number consists of 16 digits.

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.

a.

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

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

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

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 $\frac{1}{{10}^{16}}$.

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

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

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 $\frac{1}{{10}^{12}}$.

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

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

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 $\frac{1}{{10}^8}$.

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.