Can't determine whether problem involves the binomial distribution or not The question is basic: If it's sunny an average of 5 days a week, what's the probability that it'll be rainy 2 days out of 3? Assume unrealistically that the weather on a given day is independent of the other days.
I'm having a major mental block and can't see through this problem. Does the binomial distribution apply; i.e.
$\dbinom{3}{2} \left(\frac{2}{7} \right)^2 \left(\frac{5}{7} \right)^1$
or is it a simple selection without replacement:
$\frac{\dbinom{2}{2}\dbinom{5}{1}}{\dbinom{7}{3}}$
I can't convince myself one way or the other; so I solicite your advice.
 A: Is it a homework? (then you should include the appropriate tag). 
Hint: Can you come up with the probability that it rains 0 days ($P_0$), 1 day ($P_1$), and 3 days ($P_3$) for your two approaches and then check whether $$\sum_{n=0}^3 P_n =1,$$
which will give you a good indication which approach is correct...
A: I believe it's the first - binomial distribution.
What's happening is that you're repeating the same experiment (whether a day will be rainy or not) three times in a row, independently. In general, when repeating an experiment n times independently with a success rate of p, the probability of success is exactly
$\dbinom{n}{m} p^{m}(1-p)^{n-m}$
Here n=3, m=2, and $p=\frac{5}{7}$ so it should be:
$\dbinom{3}{2} \left(\frac{2}{7} \right)^2 \left(\frac{5}{7} \right)^1$
Selection without replacement implies that the experiments are dependent, which doesn't make sense here since we've assumed the weather of each day doesn't depend on the others.
A: You are right to have some hesitancy. 
If the statement about "sunny" is a general statement, meaning that the probability of sunny is $5/7$, then the binomial distribution (your first choice) is the relevant one.
But if the statement is interpreted as meaning that in a PARTICULAR week it was sunny $5$ days out of $7$, and we ask for the probability that for some given $3$ days that week it rained for exactly $2$, then the hypergeometric distribution (your second choice) is the appropriate one.
My slightly preferred interpretation of the problem would be the first (binomial) one, but it really is a matter of interpretation, and there is a good case to be made, on meteorological grounds, for the second interpretation.  It is unfortunate that, in attempts at "relevance", you are asked to thrust an overly simplified probability model on a complex situation.  
Too bad that the question did not say.  "It rained $5$ days out of $7$ during my week-long Spring break.  What is the probability that out of $3$ randomly chosen days that week, it rained on exactly $2$?" Then everything would be clear, the second interpretation would be correct.   
A: The binomial distribution apply if the experiments satisfy these criteria:


*

*The experiments are independent.

*Each experiment has two outcomes: success/failure.

*The probability for success in each experiment is the same.

