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.