Molecule collision probability 
In a time $dt$ , our molecule will sweep out a volume $\sigma vdt$.If another molecule happens to lick inside this volume, there will be a collision.With $n$ molecules per unit volume, the probability of a collision in time $dt$ is therefore $n\sigma vdt$. Let us define $P(t)$ as follows :

In my opinion, now that $n$ is the molecules per unit volume and $\sigma v\mathrm{d}t$ is the volume, then $n\sigma v \mathrm{d} t$ should be the number of molecules collided in time $ \mathrm{d} t$ instead of the probability.
Moreover, if $n \sigma v \mathrm{d} t$ is the probability, will it be greater than $1$, while probability should be a fraction not greater than $1$ ?
Can you help out that why $n\sigma v \mathrm{d} t$ is the probability instead of number?
 A: You are right, $n \sigma c \, dt$ is the number of collisions. But it is also the probability when $dt$ is small enough, as I will explain.
The collision rate is $n \sigma v$ and this is the number of collisions undergone by the molecule under consideration, per unit time. If we take a non-infinitesimal time $\Delta t$ then the number of collisions undergone by a single molecule would be
$n \sigma v \, \Delta t$ if its speed was not affected by the collisions, and this number can be larger than 1.
However, let's now define what is meant by "the probability that the molecule has a collision in some small time interval $dt$" in the limit where $dt \rightarrow 0$. Using a frequentist definition of probability, which is convenient here, we can imagine doing a large number $N$ of trials. In each trial a molecule moves for the time $dt$. We count the total number of collisions. This is $N n \sigma c \, dt$.
Next we argue that in the limit of small $dt$, in each trial there was either no collision or one collision, but not two or more collisions. Therefore in this limit the number of trials in which there was a collision is also $N n \sigma c \, dt$.
Finally we divide this by the number of trials $N$ and thus get the probability that there is a collision in any one trial:
$$
P = n \sigma c \, dt.
$$
