Why Does Persistence Pay? The Mathematical Proof

We’ve all heard it a million times that “persistence pays.”
We’ve seen countless examples of people succeeding in something just because they’ve kept trying at it until… SHAZAM! Something gives away, an invisible lever in the sky flips over, and the obstacles disappear like magic. We use the phrase “lucky break” to describe the mysterious process.
The reverse is also true: people who do not “push their luck” over and over again end up leaving the field and switching to something else. The danger is of course no matter what their new field is the same “persistence pays” challenge awaits them over on the other side of the fence as well. And sooner or later they have to face the facts and “push it” against all odds…
Well, there I said it: “against all odds…” People use this expression frequently to describe the difficulties they meet in life. But does it really make sense?
Actually, I’d like to argue that, the more you try something the more the odds start working FOR you, not against you. And that’s why “persistence really pays.”
Moreover, I think I can prove that to you mathematically…



The basis of my assertion is a simple well-known probability formula first developed by the French mathematician and philosopher Blaise Pascal (1623-1662).

Probability of A happening = 1 – (Probability of A NOT happening)

It’s as simple as that. A shorter form is: Prob A = 1 – (Prob NOT A)

And for INDEPENDENT events, Prob (A or B) = 1 – (Prob-NOT-A x Prob-NOT-B)

By “independent events” I mean things that do not have any impact on each other. For example, my question to my son and his response are not independent events. Clearly, the question conditions the answer. But I guess I can safely say that my question and an earthquake in China are independent events (although I know quantum physicists would disagree with that).
So, going back to proving mathematically that “persistence pays”…


Let’s imagine you’re applying for a number of job offers as a technical writer.
Let’s again assume that your chances of getting accepted for any of the job offers is a paltry 10%. So your chances of NOT getting accepted to any of these positions is a whopping 90% (or 0.9 in terms of the probability theory).
So let’s see how you can improve your chances of getting ANY of these job offers simply by persisting and applying to one offer after another:

Applying to 1 job: Probability of Getting Accepted to Job A = 1 – 0.9 = 0.1, or 10%
Applying to 3 jobs: P (A or B or C) = 1 – (0.9 x 0.9 x 0.9) = 1 – 0.729 = 0.271, or 27.1%
Applying to 6 jobs: P (A or B or C or D or E or F) = 1 – (0.9 raised to the power of 6) = 1 – 0.531 = 0.469, or 46.9%
Applying to 10 jobs: P (A or B or …) = 1 – (0.9 raised to the power of 10) = 1 – 0.348 = 0.652, or 65.2%
Applying to 20 jobs: P (A or B or …) = 1 – (0.9 raised to the power of 20) = 1 – 0.121 = 0.879, or 87.9%

Now you can see why persistence really pays! Even though your chances of getting a job is only 10% without persisting, if you have the stamina of hearing 19 more rejections your chances of finding a job rises almost to 90%!
If you do the math by starting with a higher chance of success (with let’s say 20% chance of getting accepted to any job) or with a higher number of applications you’ll see that your probability of getting a job approach 95-to-99% rapidly. That’s why it makes sense from a mathematical point of view to keep trying.
Reason and mathematics help. And persistence helps even more in life. That’s how you create your own “lucky breaks.”

P.S. Are events always “independent”?

Of course not. For example, in the above example, the outcome of a job application can (and do) affect the outcome of the next job application. How? Here are a few possibilities:
1] An applicant gets tired after so many job applications and gives up. Or she shows up at a job interview in a bad mood due to the memories of the previous one. That’s how one application can have an impact on the next one. In that case it would be wrong to assume that the “events” are independent of one another.
2] The more your resume circulates around the less offers you may get if the resume becomes “shop worn.” Sometimes people assume that just because you’ve been searching for a job for so long something must be wrong with your credentials or background. In that case, again, it would not be correct to use the above formula since the “events” would not be truly independent.
The more reasons to suspect that the events in question are not independent, the less appropriate it would be to use the above probability formula.