Question

A thief steals an ATM card and must randomly guess the correct four​-digit pin code from a $8$-key keypad, repetition of digits is allowed. What is the probability of a correct guess on the first​ try?

Answer 1

Step 1: Find the probability of guessing each digit correct.

The total number of keys on the keypad is $8$.

The pin is of $4$ digits.

We know that, the probability of an event is $P\left(E\right)=\frac{\mathrm{Possible}\mathrm{outcomes}}{\mathrm{Total}\mathrm{outcomes}}$

The probability of guessing the first digit right is $\frac{1}{8}$,

The probability of guessing the second digit right is $\frac{1}{8}$,

The probability of guessing the third digit right is $\frac{1}{8}$,

The probability of guessing the fourth digit right is $\frac{1}{8}$,

Step 2: Find the probability of a correct guess of the pin.

To obtain the probability of a correct guess on the first​ try $P\left(E\right)$, using the multiplication rule of probability.

$\begin{array}{rcl}P\left(E\right)& =& \frac{1}{8}\times \frac{1}{8}\times \frac{1}{8}\times \frac{1}{8}\\ & \Rightarrow & P\left(E\right)=\frac{1}{8^4}\\ & \Rightarrow & P\left(E\right)=\frac{1}{4096}\\ \therefore P\left(E\right)& =& 0.00024\end{array}$

Hence, the probability of a correct guess on the first​ try is $0.00024$.