Q.6.17 - A First Course in Probability

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 $X_{1 }and X_3$?

Step-by-Step Solution

Answer

The probability that $X_2$ lies between $X_1 and X_3$ is $\frac{1}{3}$

Step 1: To find

The probability that$X_2$lies between $X_1 and X_3$

Step 2: 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 $X_{2 } lies between X_1 and X_3 in 2 arrangements$

Therefore $$\begin{gathered} X_{2 } lies between X_{1}and X_3 =\frac{2}{6} \\ =\frac{1}{3} \end{gathered}$$