Probabilities above $1$

Is it okay for a probability to be above one, where all probabilities above one are equivalent to a probability of one? If not, is it okay to write something like this:

$$P(Q) = \begin{cases} \displaystyle \frac ab, & \text{if} \ \ \displaystyle \frac ab \le 1 \\[2ex] 1, &\text{if} \ \ \displaystyle \frac ab > 1 \end{cases} $$

EDIT:

In a comment, I said:

I think that a probability above 1 has been established as non-sensical in the comments, or non-standard (pretty sure my theorem works within standard probability theory). As such, my question defers to whether the given expression is allowed.

This was answered by Ethan Bolker, and I was planning on accepting his answer. However, the nature of $Q$ seems to be of relevance, hence my reluctance to do so, and the previous edit I made, shown below:

I posted the theorem I was working on here. In it you can see what caused this conundrum. According to Ethan Bolker's answer, there was no problem in using the piecewise expression, so I did that.

A probability is a special case of an expected value. Say you're picking balls from an urn and looking to see whether they are red. The probability of a red ball is exactly the expected number of red balls per pick.

If you are picking two balls at a time, you can still speak of the expected number of red balls per pick and it could be as high as $2$. That's no longer a probability, but it's closely related to one; you can see that the two situations are very similar.

You didn't tell us what your original question was, that provided the context for your “probability greater than 1”. It's possible that what you really have is an expected value and you are confusing it with a probability.

For a better answer, try asking a more specific question.

In a comment (that should really be part of the question you clarify:

As such, my question defers to whether the given expression is allowed.

There is no mathematical problem at all with the piecewise expression you are using to define a probability.

Whether that cutoff makes sense when $a>b$ depends on what $a$ and $b$ mean in your context.