March Madness season is back, and with it are disappointments of a busted bracket. I stumbled upon an article (from the Weather Channel — what?!) about the probabilities of getting a perfect bracket: 1 in 9,223,372,036,854,775,808. But just *how* small is that? Let’s do some math to find out!

Suppose everyone in the world, population , fills out a random bracket at every time interval , at the same time. Let the probability of a person getting the perfect bracket be . The probability that nobody gets the bracket right on any time interval is

,

and thus the probability that a person will get the bracket at any time interval is

.

The probability that someone on Earth gets the bracket right for the first time at the *k*th time interval is the probability that nobody gets it right in the first intervals and that someone gets it right on the *k*th interval. Let’s denote a random variable corresponding to the interval that someone on Earth finally gets the bracket right. Putting this all together, we have

.

The expected interval of someone getting it right, i.e. the expectation value of is calculated as,

.

To simplify this problem, let

,

or

,

such that

.

To perform the summation we observe that

.

So the expectation value calculation becomes,

,

where the derivative has been taken out of the summation. The quantity in the parenthesis is just a sum of an infinite geometric series with ratio between successive terms equal to , so it can be performed easily to yield:

.

After taking the derivative, and subsituting back for we get

.

Now theoretically that *does* it, but we can make our arithmetic easier by observing that . Then, using the binomial expansion

,

we obtain

.

So now to answer our question. From the weather.com article linked earlier, we have

.

According to Wolfram Alpha, the estimated world population in 2013 was

.

Doing the math, we get the .

If everyone in the world filled out a bracket every second, we would expect about** 40 years** before someone gets it right.

If everyone in the world filled out a bracket every year, we would expect **1.29 billion years** before someone gets it right, or about **1/10 the age of the universe**.

Good luck with that!

EDIT: Just realized, we were not *exactly* correct in saying alone justifies using the binomial expansion approximation, as . However, that the product *does* fully justify our approximation. The author regrets this error.