# Are There Mathematically More Reasons to Keep—or to Quit—a Job?

Are there more potential (as opposed to real, current) reasons to leave a job than potential reasons to keep it?

Or is it the other way around? You may think that is either an unanswerable question, that the answer is “no” or that it’s a matter of specific circumstances, not in any way a theoretical mathematical question. After all, “it all depends”—right?

But compare the question of whether there are more ways to do a job right than to do it wrong, for example, on an assembly line. Do you want to say here, too, that it’s unanswerable, that it depends, or “no”?

Think again: There’s only one way to get a sequenced job right: each step, e.g., assembly of an automobile, has to be performed as specified; otherwise the final result will be flawed, i.e., wrong.

**Assembly-Line Math**

So, if there are, say, 500 steps in the assembly of some car, there are 500 ways the job of assembling one can go wrong, if there is only one mistake, but only one way the complete assembly can be done right (i.e., fall within specified tolerances for each step).

When the possibility of more than one assembly-line mistake in one completed assembly is considered, the maximum possible number of ways to fail becomes astronomical, viz., 2^{500}-1. This reveals the error in the simplistic “coin toss” analysis that says that because the assembly is either right or wrong, there must be an equal number of ways of being successful or unsuccessful (namely one each).

So, what about the number of ways in which keeping or leaving a job is the right thing to do? Can these two numbers be calculated and compared in the same way as the ways of getting an automobile assembly right or wrong?

**Necessary vs. Sufficient Conditions**

As a first step to answering this, let’s distinguish *necessary conditions* for quitting a job from *sufficient conditions* for doing so. A necessary condition for quitting or otherwise leaving a job can, for a given individual, include things like a low salary (which is not the same thing as an unacceptable salary, which, by definition, would be a sufficient condition for quitting).

It’s only a necessary condition for quitting, if it’s a factor, but not enough to make you walk, e.g., when you also love what you do enough to stay, unless something else goes seriously awry with the job. Then that low salary becomes part of an unacceptable job “package”.

**2 ^{N}-1 vs. 1**

On the other hand, suppose you’ve decided that a pay cut, a better salary offer elsewhere, a change for the worse in management, a serious health problem, a flood or other natural disaster that wipes out your company, an irresistible urge to travel, a dangerous office love affair, a painful increase in your work load, etc., would individually be sufficient, i.e., enough of a reason, for you to quit or otherwise not continue working. Count up all of these and call the total “N”.

If enumerating these seems impossible, impractical or subject to frequent variation, just imagine that the list is nonetheless always finite and still call the total “N”. Among these conditions might be an *unacceptably* low salary.

Allow for the possibility that more than one of these may happen at the same time, e.g., your salary is unacceptably cut and you have an irresistible urge to travel. This means that there are theoretically and potentially 2^{N}-1 sufficient reasons to quit your job. On this analysis, how many reasons are there for you to stay on the job? Just one—namely, because not one of these reasons to quit materializes. There’s only one way that can happen.

Why 2^{N}-1 reasons? Consider a simple example: You are betting that in rolling one die six times you will guess correctly each and every time. How many ways can you succeed? Only one—by guessing all correctly. How many ways can you fail? 2^{6}-1, which equals 2x2x2x2x2x2 -1, or 63 ways to make at least one wrong guess. The “2” represents the number of possible outcomes for each roll, namely, right or wrong.

The same analysis applies to tossing three coins: There’s only one way to guess all three correctly, but seven ways to make at least one mistake (2x2x2 -1).

So, on this analysis, there is only one potential reason to keep your job—because not one of the sufficient conditions for leaving the job has been triggered; but, as shown, there are 2^{N}-1 potential reasons to quit. This analysis can be generalized: For any given job, there will always be more potential reasons to quit the job than to keep it—if, and this is a big if, the reasons for *quitting *the job are formulated as *sufficient* conditions. Specifically, there will be only one reason to keep the job and 2^{N}-1 potential reasons to quit.

**Enough to Leave vs. Enough to Stay**

What happens when the analysis shifts from enumerating the sufficient conditions for *leaving* to sufficient conditions for *staying*? This requires (however unrealistically) imagining that there are N individual reasons for staying, each sufficient to keep you at the job, e.g., an astronomical salary or fabulous creative challenges. This means a kind of “over-thrill” (corresponding to job-nuking reasons “over-kill”) in having multiple conditions or circumstances sufficient to make you *stay* on the job.

But, hold on!—Although there are N individual sufficient reasons for staying, they can potentially or in fact exist in combinations of each other, e.g., that fabulous salary* and *the unique creative challenges. This means there are 2^{N}-1 potential combinations of reasons *not* to quit. OK. Well, then, how many potential reasons, on this analysis, are there for leaving the job?

Just one: when* none* of the sufficient reasons for staying applies (or, strategically—if not logically—equivalently, when one or more necessary conditions are not met). Just one? Yep.

Yes, because if none of the sufficient conditions for staying at a job (or for anything else) is met, it means that not enough necessary conditions have been met to warrant staying at the job. It’s as though the “critical mass”, the logical threshold for staying has not been met, just as the omission of water from the inventory of necessities for camping means insufficient, failed preparation.

**Confusing?**

Now this is confusing, right? On the first analysis, the number of potential reasons to leave a job dwarfs the one reason to stay; but, on the second analysis, it’s the reverse: the number of potential reasons to stay dwarfs the one potential reason to leave! How is that possible and which analysis counts more?

The answer is that *it depends on your focus*: If your focus is on what is enough to make you *leave*, the number of reasons for leaving will vastly outnumber the one reason to stay, namely, that not one of the reasons to leave has been triggered; but, if your focus is on what is enough to make you *stay*, the number of reasons to stay will dwarf the one reason to leave, namely, that none of the sufficient conditions for leaving has materialized.

(To complicate things even more, the whole analytical approach can be shifted from itemization of *sufficient* conditions for staying or leaving to *necessary* conditions, as parenthetically hinted at above. I leave this as an exercise for the motivated reader, but suggest that the results will mirror the calculations and conclusions presented here.)

**How Your Hunch Was Right and Wrong**

So, in a sense, if your original hunch in answering the title question was that “it depends”, you were right—but also a little wrong, i.e., right, but probably not for the reason you imagined (since it’s unlikely that your hunch was based on the foregoing mathematics and logic). Most who would say “it depends” probably mean “depends on circumstances”, rather than, as this analysis suggests, “depends on focus”.

Another takeaway from all of this is that, in evaluating your own job or a job to offer a candidate, you should take care to distinguish and identify the conditions that would be necessary for the candidate to accept or decline, keep or leave a job, from those that would be sufficient to do so.

For example, an unsatisfactory compensation package may be only a necessary, not a sufficient, condition for an individual’s leaving a job or rejecting an offer.

The challenge, as a recruiter or as an employee, is to determine whether that unsatisfactory package is also unacceptable and in how many ways (even if only to forecast the scope of negotiations required to save the deal).

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