Basics Of The Binomial Distribution

Even if you don’t know the binomial distribution by name, and never took an advanced college statistics class, you innately understand it. Really, you do. It’s a way of assessing the probability of a discrete event either happening, or failing to happen. And it’s got plenty of applications in finance. Here’s how it works:

You start by attempting something – coin flips, free throws, roulette wheel spins, whatever. The only qualification is that the something in question must have exactly two possible outcomes. Success or failure, that’s it. (Yes, a roulette wheel has 38 possible outcomes. But from a bettor’s standpoint, there are only two. You’re either going to win, or lose.)

We’ll use free throws for our example, because they’re a little more interesting than the exact and immutable 50% chance of a coin landing heads. Say you’re Dirk Nowitzki of the Dallas Mavericks, who hit 89.8% of his free throws in the 2017–2018 season. We’ll call it 90% for our purposes. If you were to put him at the line right now, what are the chances of him hitting (at least) nine out of 10?

No, they’re not 100%. Nor are they 90%.

They’re 74%, believe it or not. Here’s the formula. We’re all adults here, there’s no need to be scared of exponents and Greek letters:

 n is the number of attempts. In this case, 10.

 i is the number of successes, which is either nine or 10. We’ll calculate the probability for each, then add them.

p is the probability of success of each individual event, which is 0.9.

The chance of reaching the target, i.e. the binomial distribution of successes and failures, is this:

i = 0 k ( n i ) p i ( 1 p ) n i \begin{aligned}&\sum^k_{i=0}\left(\begin{matrix}n\\i\end{matrix}\right)p^i(1-p)^{n-i}\end{aligned} i=0k(ni)pi(1p)ni

Remedial math notation, if you need the terms in that expression broken down further:

( n i ) = n ! ( n i ) ! i ! \begin{aligned}&\left(\begin{matrix}n\\i\end{matrix}\right)=\frac{n!}{(n-i)!i!}\end{aligned} (ni)=(ni)!i!n!

That’s the “binomial” in binomial distribution: i.e., two terms. We’re interested not just in the number of successes, nor just the number of attempts, but in both. Each is useless to us without the other.

More remedial math notation: ! is factorial: multiplying a positive integer by every smaller positive integer. For instance,

5 ! = 5   ×   4   ×   3 ×   2 5!=5\ \times\ 4\ \times\ 3\times\ 2 5!=5 × 4 × 3× 2

Plug the numbers in, remembering that we have to solve for both 9 out of 10 free throws and 10 out of 10, and we get

( 10 ! 9 ! 1 ! × . 9 . 9 × . 1 . 1 ) + ( 10 ! 10 ! × . 9 1 × . 1 0 ) \left(\frac{10!}{9!1!}\times.9^{.9}\times.1^{.1}\right)+\left(\frac{10!}{10!}\times.9^1\times.1^0\right) (9!1!10!×.9.9×.1.1)+(10!10!×.91×.10)

= 0.387420489 (which is the chance of hitting nine) + 0.3486784401 (the chance of hitting all ten)

= 0.736098929

This is the cumulative distribution, as opposed to the mere probability distribution. The cumulative distribution is the sum of multiple probability distributions (in our case, that’d be two.) The cumulative distribution calculates the chance of hitting a range of values—here, nine or 10 out of 10 free throws—instead of a single value. When we ask what the chances of Nowitzki hitting nine out of 10 are, it should be understood that we mean “nine or better out of 10,” not “exactly nine out of 10.”

If you want to figure out the binomial distribution function for a particular series of events, you don’t have to calculate it yourself. The helpful folks at Stat Trek have a binomial calculator that’ll do the work for you. All you have to do is supply the n, i and p values.

So what does this have to do with finance? More than you might think. Let’s say you’re a bank, a lender, who knows to within three decimal places the likelihood of a particular borrower defaulting. What are the chances of so many borrowers defaulting that they’d render the bank insolvent? Once you use the cumulative binomial distribution function to calculate that number, you have a better idea of how to price insurance, and ultimately how much money to loan and how much to keep in reserve. 

Ever wonder how options’ initial prices are determined? Same thing, sort of. If a volatile underlying stock has a p chance of hitting a particular price, you can look at how the stock moves over a series of n periods to determine what price the options ought to sell at.

Applying the binomial distribution function to finance gives some surprising, if not completely counterintuitive results; much like the chance of a 90% free-throw shooter hitting 90% of his free throws being something less than 90%. Assume you’ve got a security that has as much chance of a 20% gain as it does a 20% loss. If the security’s price were to fall 20%, what are the chances of it rebounding to its initial level? Remember that a simple corresponding gain of 20% won’t cut it: A stock that falls 20% and then gains 20% will still be down 4%. Keep alternating 20% falls and gains, and eventually the stock will be worthless.

The Bottom Line

Analysts with a grasp of the binomial distribution have an additional quality set of tools at hand when determining pricing, assessing risk, and avoiding the unpleasant results than can accrue from insufficient preparation. When you understand the binomial distribution and its often surprising results, you’ll be well ahead of the masses.

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  1. "NBA Advanced Stats: Dirk Nowitzki." Accessed May 7, 2021.

  2. Stat Trek. "Binomial Probability Calculator." Accessed May 7, 2021.

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