Question

Three points $X_1,X_2,X_3$ are selected at random on a line $L$. What is the probability that$X_2$ lies between $X_{1}and{X}_3$?

Short answer

The probability that $X_2$lies between $X_{1}and{X}_3$is$\frac{1}{3}$

Step-by-step solution

To find

The probability that$X_2$lies between $X_{1}and{X}_3$

Explanation

Three points to consider On a line $L,X_1,X_2,X_3$are chosen at random.

The number of ways to arrange n items all at once$=n!.$

Total number of point$=3$As a result, the number of ways to arrange three points on a line is$$\begin{gathered} =3! \\ =6 \end{gathered}$$

The possible arrangements are:

$\begin{array}{r}{X}_1{X}_2{X}_3\\ X_1{X}_3{X}_2\\ X_2{X}_1{X}_3\\ X_2{X}_3{X}_1\\ X_3{X}_1{X}_2\\ X_3{X}_2{X}_1\end{array}$

Out of these $X_{2}liesbetween{X}_{1}and{X}_{3}in2arrangements$

Therefore $$\begin{gathered} X_{2}liesbetween{X}_{1}and{X}_3=\frac{2}{6} \\ =\frac{1}{3} \end{gathered}$$