How to select distribution? — Binomial, Poisson, ... How do I go about finding which distribution I need to use for my exercise? 
I have the following exercise:

Compute the probability that within a group of 5 students exactly two
  are born on a Sunday.

What gives me a hint on what probability distribution that is?

The Guide from the previous answer is helpful. It should help you pinpoint the exact distribution to choose. 
Sometimes the problem might not be easily convertible to a distribution. In those cases you can always go back to thinking in terms of basic probabilities. 
How many total ways can the student birthday be arranged? Number of days to the power of number of students =  $7^5$ 
How many ways can we arrange the students to satisfy the condition? Choose 2 students from the 5. They are born on a Sunday. The rest isn't. This leads to ${5 \choose 2}$ * $1^2$ * $6^3$
If you divide the two you will get ${5 \choose 2} * {\frac 1 7}^2 * {\frac 6 7}^3$
Notice that it's the same result if you were to model it using binomial distribution with p = 1/7, n = 5. So at the end of the day, if you feel stuck you can always roll back to basics. 
Hope it helps
