Question

Consider the set of all permutations of the numbers 1, 2, 3. If you select a permutationat random, what is the probability that the number 2 is in the middle position?In the first position? Do your answers suggest a simple way of answering the same questions for the set of all permutations of the numbers 1 to 7?

Short answer

The probability that the number 2 is in the middle position each is$\frac{1}{3}$ .

The probability that the number 2 is in the first position each is $\frac{1}{3}$.

For numbers 1 to 7, the probability will be $\frac{1}{7}$.

Step-by-step solution

Definition of Permutation

A specific way or order in which a particular set of objects, numbers, or things is arranged is defined as permutation. With help of this concept, the number of arrangements can be determined.

Creation ofsample space for experiment for the permutation

When the numbers 1,2 and 3 are arranged, there are total 6 possible combinations. So, the sample space is expressed as follows.

$123,132, 213, 231, 312, 321$

Determination ofprobability that the number 2 is in the middle position

It is observed that each point of the sample space has an equal probability of $\frac{1}{6}$.

Find the probability that the number 2 is in the middle position by adding the probabilities of the points which has number 2 in the middle.

$\begin{array}{rcl}p& =& \frac{1}{6}+\frac{1}{6}\\ & =& \frac{2}{6}\\ & =& \frac{1}{3}\\ & & \end{array}$

Determinationof probability that the number 2 is in the first position

Find the probability that the number 2 is in the first position by adding the probabilities of the points which has number 2 in the middle.

$\begin{array}{rcl}p& =& \frac{1}{6}+\frac{1}{6}\\ & =& \frac{2}{6}\\ & =& \frac{1}{3}\\ & & \end{array}$

Thus, the probability that the number 2 is in the middle position each is and the probability that the number 2 is in the first position each is $\frac{1}{3}$.

It can be observed that when the position of any number is fixed, the probability is reciprocal of total number of digits given. And, thus for numbers 1 to 7, the probability will be $\frac{1}{7}$.