Determining whether a given function can be considered as a probability function Recently I was given the following problem.

The sample space $\Omega = \{ 1, 2, 3, 4 \}$ and the function P is defined as follows: 

$$P(\emptyset) = 0, P(\Omega) = 1, P(\{1,2\}) = \frac12, P(\{3,4\}) = \frac12, P(\{3\}) = \frac13, P(\{4\}) = \frac16 , P(\{1,2,3\}) = \frac56, P(\{1,2,4\}) = \frac23.$$

The question is whether $P$ can be considered a probability function.

My issue with this problem is that I am not sure whether it is definitely possible to answer the question. On the one hand, if P is a probability function, then for each two mutually exclusive events the sum of their probabilities should be $1$. In this case, $\{1\}$ and $\{2,3,4\}$ are mutually exclusive events, but $P(\{1\}) + P(\{2,3,4\})$ is not known. On the other hand, there is not enough information to claim the opposite. It seems to me that the question itself is incorrect, but I am not quite sure because I do not have enough experience in probability theory.

So, is $P$ a probability function?
When I took probability 1, there was a triple $(\Omega, P, \mathcal F)$ consisting of a sample space, a probability measure, and a collection of events, subsets of $\Omega$ to which probabilities are assigned.  The domain of $P$ is  $\mathcal F$.

So I'd give a different answer to this question.  The set of events $\mathcal F$ has not been specified, so we can choose it.  If we take $\mathcal F$ to be all subsets of $\Omega$, then Jeremy's answer is correct: at best $P$ is incompletely specified.  But if we take $\mathcal F$ to be generated by just the subsets $\{1,2\}, \{3\}, \{4\}$, then $P$ is a probability on that $\mathcal F$.  If asked, in this case, what $P(\{1\})$ was,  we could answer: ha-ha, $\{1\}$ is not measurable, is not an event, is not in $\mathcal F$, and it doesn't have (and doesn't need) a probability.
This seems like a trick question. Much information is provided about a possible probability function, $P$. Since the sample space is $\omega = \{1, 2, 3, 4\}$, the probability function must describe the probability of each element of the sample space. Events $1$ and $2$ are defined in three of equalaties together. Since the probability of events $1$ and $2$ aren't defined individually and they are part of the sample space this probability function $P$ is incomplete and not fully defined.

$P$ is not a probability function.