In academia, it's all about getting grants that allow you to pay your
expenses while performing the research you really like. However, money
has always been tight everywhere, and as such we often do not get the
grant we aspired to. Getting rejected from a grant is not a great
feeling, and it may feel arbitrary getting rejected with something you
poured weeks of your life into.
Sometimes it helps to take out some hard math to figure out the big
question: Were you truly unlucky, or is your time to shine still coming?
On this page, you can get a feeling for how competitive a grant is by
providing the most recent acceptance rate of the grant, and discovering
how often you would have to apply to the grant before your chances are
starting to look good in getting that grant.
Grant-o-Graph
This graph plots the cumulative probability of you getting this grant
based on the number of attempts you already tried. It highlights four
special points, in this order:
First the attempt at which the initial chances of receiving the grant will have doubled.
Second the attempt at which your chances to receive the grant are above 50 %.
Third the attempt at which you have tried so often, the chance of getting the grant are 99 %.
Lastly, and optionally, marked in a different color is the attempt at which you are sitting right now.
As you can see, the probability follows a logarithmic function. In your
first few attempts, the chances of receiving the grant will rise sharply,
but there is a point of diminishing returns. Essentially, this means that
if you still don't get the grant after that point despite trying,
the universe may simply be against you.
The Math
The stated problem is probability based and relatively straight forward.
You have an empirical (observed) acceptance rate for grants (i.e., by
looking at previous grant decisions). What you want to know is how often
you'd need to try until the chance of you finally landing at least one
of them is in a (for you) comfortable territory. (Mind you, we're
talking about probabilities, so even with a 99 % chance, the
universe can still be cruel to you and simply not make it happen
regardless of how often you try.)
There are three assumptions behind this math:
Your submitted proposals are all good enough to be actually considered for funding.
Subsequent grant decisions are independent of previous decisions.
The general acceptance rate remains stable over time.
Formally, we are talking about a variation of a coin toss.
We can think of grant decisions as independent tosses of a biased coin,
where heads (success) is less likely than tails (failure). The more you
flip the coin, the more likely it will become that the coin will at some
point land on heads, because even though each coin toss is independent of
the previous ones, you only need to succeed once. The question therefore
is: Given a known probability of landing heads, how many times would you
need to flip the coin such that the chances of landing heads are above a
certain threshold?
This is difficult to estimate, because any of your attempts may be
successful. Instead, the insight here is that you can also calculate the
reverse probability, that is, the probability of never
succeeding, which is just the probability of
(1-p) * (1-p) * … * (1-p), or in short:
(1-p)n where 1-p is the
probability of failure, and n is the number of coin tosses.
That n is what you want to know.
To get a sense of how often we would have to try, we need another
variable, however: What is a chance of getting the grant we feel
comfortable with? That's P (a capital p). For
example, maybe we want to be really, really certain that we can succeed,
in which case we should choose a high P of, say, 90 %.
Or, we may want to be just in the territory of "more likely than not,"
in which case we would choose P = 50 %.
This leads to a very simple inequality: 1-P < (1-p)^n.
While this is not an equation, but an inequality, solving for
n is straight-forward. We just need to use a logarithm to
pull the wanted quantity out of its exponent position.
To calculate n with your chosen value of
and, say, a cumulative
probability of , this is how we
solve it:
All in all, you would need attempts until
your chances of getting the grant are above the desired cumulative
probability.
Motivation
As mentioned in the beginning, competing for grants is a game of
attrition. After yet another lost race for continued employment, I
decided to pour my frustration with the russian roulette we call grant
season into a productive little project – this website.
The reason I decided to approach it like this is because, after an
earlier unsuccessful application, my former supervisor said that even
though each grant decision is independent, the chances of getting a
grant will inevitably rise, because you will only need one single
successful application to go through. In other words, there is still
merit in working hard and keeping up with the grant game.
Personally, I believe that this is not how science should work, because
the ratio of good but unsuccessful to successful grant applications is
untenable. But, well, here we are.