Appropriate distribution for the given problem Red blood cell deficiency may be determined by examining a specimen of the blood under a microscope. Suppose that certain small fixed volume contains on an average $20$ red cells for normal persons. What is the probability that a specimen from a normal person will contain less that $15$ red cells ?
I thought of using the Exponential distribution for this problem since a single parameter is involved and the average number of cell is given.
Thus , $N$ denotes the number of red cells present in the specimen of a normal person. So , $N$ ~ $Exp(\lambda)$ , where $ \frac{1}{\lambda} = 20$.
Hence , $P(N < 15)$ can be easily calculated. But as far as I know exponential distribution is used whenever some 'time' related or a 'process' related thing is going on , here although I was able to get a solution  , but that doesn't seem correct. Does it ?
 A: 
Suppose that certain small fixed volume contains on an average 20 red cells ... I thought of using the Exponential distribution for this problem since a single parameter is involved and the average number of cell is given.

IMHO, your second statement doesn't quite follow from the first. It's unclear that a single parameter (specifically the mean) is involved. A single parameter (specifically the mean) is what's been recorded and given here. Given samples generated by any distribution (including those with multiple parameters), you could just state what the average happened to be.
There are two non-exclusive ways you could go. 
The first is by trying to deduce the distribution through the underlying process. For example, looking at erythropoiesis, one could start with the following model (I'm not a biologist, so the details might be wrong - this is just an example). Stem cells generate blood cells at an extremely-high fixed rate, each cell is ready in about 7 days, and lives for for a number of days that is normally distributed. Under the assumption that the blood volume is approximately constant, the density of blood cells could be approximated as normal.
For other underlying processes, I'm guessing the distribution could be that of a queuing system or a Poisson distribution.
The second is by starting with a specific distribution (or small number of distributions), and testing the hypothesis that the sample is generated by the distribution. See here how to check if a process is normal, see here how to check if the process is Poisson, and see here how to check if a process is exponential or normal. 
Once you decide which distribution best fits the data, you will probably need to estimate the parameter(s). See here, for example, how to estimate the parameters of a normal distribution.
A: Let us say ,on an average 20 red cells are present per $x$ unit volume.
Thus , if $N$ represents the number of red cells we can have : $N$ ~ $Pois(\lambda)$, where  
$\lambda = 20$ cells per $x$ unit volume.
Hence $P(N<15)$ can be calculated.
