Question

There are m persons sitting in a row. Two of them are selected at random. The probability that the two selected person are not together is
A.$\dfrac{2}{m}$
B.$\dfrac{1}{m}$
C.$\dfrac{{m(m - 1)}}{{(m + 1)(m + 2)}}$
D.$1 - \dfrac{2}{m}$

Answer 1

Hint: We will use the combination formula in the probability of finding the solution to this question. The combination formula is $\dfrac{{n!}}{{r!\left( {n - r} \right)!}}$. The probability of an event is defined as the chance of that event happening.

Complete step-by-step answer:
According to the question, we know that we have a total m number of persons sitting in a row.
The number of ways in which we can randomly select two persons at a time will be ${}^m{C_2}$.
So, the sample space will be ${}^m{C_2}$.
Number of ways to select 2 people such that they are sitting together or in consecutive positions$= m - 1$
So, the number of ways in which the two selected persons are not together$= {}^m{C_2} - (m - 1)$
Therefore, the probability that the two selected person are not together will be

$$= \dfrac{{{}^m{C_2} - (m - 1)}}{{{}^m{C_2}}} \\\ = 1 - \dfrac{{m - 1}}{{{}^m{C_2}}} \\\ = 1 - \dfrac{{m - 1}}{{\dfrac{{m!}}{{2!\left( {m - 2} \right)!}}}} \\\ = 1 - \dfrac{{m - 1}}{{\dfrac{{m(m - 1)}}{2}}} \\\ = 1 - (m - 1) \times \dfrac{2}{{m(m - 1)}} \\\ = 1 - \dfrac{2}{m} \\\$$

Hence, the required probability is $1 - \dfrac{2}{m}$.
Thus, the answer is option D.

Note: In these types of questions where we are told to make a selection, we will use the combination formula. If we are told to arrange, then we will use the permutation formula. So, we need to use the respective concept according to the question.