Chapter 9.7, Problem 48E

a.

To determine

To find: the probability of guessing the correct sequence at random.

Answer to Problem 48E

  $\frac{1}{10000}$

Explanation of Solution

Let E be the event that you guess the correct sequence at random.

Since, there is only 1 possible way that all digits are in correct order. So,

  $n\left(E\right)=1$

Since, in PIN codes, the digits can be repeated. So, there are 10 ways to choose each of the 4 digits of the sequence. So, total number of possible combinations is:

  $\begin{array}{c}n\left(S\right)={10}^4\\ =10000\end{array}$

Thus, the probability of guessing the correct sequence at random is:

  $\begin{array}{c}P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}\\ =\frac{1}{120}\end{array}$

b.

To determine

To find: the probability of guessing the correct sequence when you recall the first two digits.

Answer to Problem 48E

  $\frac{1}{100}$

Explanation of Solution

Let E be the event that you guess the correct sequence when you recall the first two digits.

Since, there is only 1 possible way that all digits are in correct order. So,

  $n\left(E\right)=1$

Since, in PIN codes, the digits can be repeated. So, there are 10 ways to choose each digits of the sequence. But of the 4 digits, two are already known. So, only two digits need to be guessed. So, the total possible number of combinations is:

  $\begin{array}{c}n\left(S\right)={10}^2\\ =100\end{array}$

Thus, the probability of guessing the correct sequence when you recall the first two digits is:

  $\begin{array}{c}P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}\\ =\frac{1}{100}\end{array}$