Question

Two boys A and B find the jumble of n ropes lying on the floor. Each takes hold of one loose end and randomly. If the probability that they are both holding the same rope is $\dfrac{1}{101}$ then the number of ropes is equal to.
(a) 101
(b) 100
(c) 51
(d) 50

Answer 1

Hint: Focus on the point that a rope has two ends. Try to find the number of ways for the second boy to select the other end of the rope that the first boy has selected.

Complete step-by-step answer:
Before moving to the question, let us talk about probability.
In simple words, we can say that probability is the possibility of an event to occur.
We can define probability in mathematical form as $=\dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$ .

To start with the question, let us first try to interpret the data given in the question. The question says that there is a jumble of n ropes lying on the floor. We know each rope will have two ends. Therefore, the total number of ends of rope lying on the floor = 2n .
It is given that person A selects one end of this 2n ends of rope lying on the floor. So, the free ends of rope left on the floor = 2n – 1 .

Now person B is to select a rope. So, the total possible outcomes for the selection of an end of a rope by B is $^{2n-1}{{C}_{1}}$ . Among this $^{2n-1}{{C}_{1}}$ outcomes the favourable outcome is only 1 i.e., when B selects the opposite end of the rope selected by A.

So, the probability of the two persons selecting the same rope is: $\text{Probability}=\dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$
$\Rightarrow \text{Probability}=\dfrac{1}{^{2n-1}{{C}_{1}}}$
$\Rightarrow \text{Probability}=\dfrac{1}{2n-1}.............(i)$
Now it is given that the probability of both persons selecting the same rope is $\dfrac{1}{101}$ . On putting the probability in equation (i), we get:
$\dfrac{1}{101}=\dfrac{1}{2n-1}$
On cross-multiplication, we get:
$2n-1=101$
$\Rightarrow 2n=102$
$\therefore n=51$
So, 51 ropes are lying on the floor.
Hence, our answer is option (c) 51 .

Note: It is preferred that while solving a question related to probability, we should always cross-check the possibilities, as there is a high chance that we might miss some or have included some extra or repeated outcomes. In the above question, do not forget to subtract one from the total outcomes as the first person has already selected one end.