Q:

Identifying the Sample space

I always have doubt about identifying sample space for events. Do they vary with the events or are they always fixed?
For eg:Tossing two coins one after another
Sample Space: $\{HH,HT,TH,TT\}$.
This was fairly easy because I could kinda relate this but how do I identify sample space for difficult questions.
For e.g., $3$ persons work independently on a problem.first person has a probability of $1/5$ to solve it,second $1/4$ and third $1/5$. What is the probability that they solve the question independently.
Sample Space-The Sample space $c$ be an the combination of no of elements of the  persons solving and not solving and all solving. Had to think for $2$ - $3$ minutes, is it right? Also does the question play any role in deciding sample space or is it fixed?
My point is that identifying sample space becomes difficult whenever the situation becomes complex so is there any guidelines to follow for identifying in Sample space.Like I had to think hard for the previous question about including the combination of elements of solving and not solving of the three person.Also how do we choose these abstract elements e.g. "solved " or "not solved".
Thanks in advance

A:

In order to apply Sample Space Theory to word problems concerning probabilities/odds, you have to do two things:

Describe the problem as a random experiment
Select a probability model

There may be more than one valid way to do this, so you can't just plug into a formula or apply a simple technique - you have to understand what you are doing. One might set things up with a 'collapsed' sample space, while another approach may give a larger space, requiring more 'counting up stuff', but they both give the same answers.

For your specific problem, I suspect you are looking for the probability that the problem gets solved by at least one person.
This is a problem that can be solved in more than one way. Below we solve it in the most complicated way. But it doesn't matter, you simply 'turn the math crank' to get the answer.
$\quad A_1 = \{1,F_1,F_2,F_3,F_4\}$
$\quad A_2 = \{1,F_1,F_2,F_3\}$
$\quad A_3 = \{1,F_1,F_2,F_3,F_4\}$
Experiment: Ask the first person if he has the solution, and he responds with an outcome listed in $A_1$. Ask the second person .... Ask the third person .... Record the answers given by the problem solvers.
Sample Space $S = A_1 \times A_2 \times A_3 $
Model: All outcomes are equally likely.
$S$ has $100$ outcomes. Let $E$ be the event that at least one of the three coordinates in $S$ is equal to $1$ (for example, $(1,F_2,F_2)$ or $(1,F_1,1)$). The set $E$ has $52$ elements so the probability that the problem gets solved is $0.52$.
Of course it looks weird to have the first person say that the outcome of his work is $F_1$ - the failures can be lumped together as one outcome. But I set this up using the discrete uniform distribution.
Exercise 1: Try using a different model with a smaller sample space. Also, try using the 'complement trick' -  What is the probability that nobody solves the problem?
Exercise 2: Get the same answer by multiplying the probabilities of independent events.

The OP writes

I always have doubt about identifying sample space for events. Do they
  vary with the events or are they always fixed?

You identify a sample space to describe all the outcomes of the random experiment. The sample space is the event that occurs with probability = 1. All other events are subsets of the (universal) sample space.

