# Basic knowledge of binomial distribution

Even if you don’t know the binomial distribution of the name, and you haven’t taken a statistical course in a higher university, you know it by nature. Really, you know. It is a method of assessing the probability of occurrence or non-occurrence of discrete events. It has many applications in the financial field. This is how it works:

You start by trying a few things-toss a coin, free throws, spin roulette, etc. The only restriction is that the thing in question must have exactly two possible outcomes. Success or failure, nothing more. (Yes, roulette has 38 possible outcomes. But from the point of view of the bettor, there are only two. You either win or lose.)

We will use free throws in our example because they are more interesting than the accurate and immutable 50% chance of a coin landing head-on. Suppose you are Dirk Nowitzki of the Dallas Mavericks. His free throw percentage was 89.8% during the 2017-2018 season. For our purposes, we call it 90%. If you put him on the line now, how likely is he to hit (at least) nine out of ten?

No, they are not 100%. They are not 90% either.

They are 74%, believe it or not. This is the formula. We are all grown-ups, there is no need to be afraid of indexes and Greek letters:

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

A generation Is the number of successes, which can be 9 or 10. We will calculate the probability of each and then add them together.

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

The chance of reaching the goal, the binomial distribution of success and failure, is as follows:

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

A generation=gram(nA generation)pA generation(1p)nA generation

Remedy the mathematical notation, if you need to further decompose the terms in the expression:

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

(nA generation)=(nA generation)!A generation!n!

This is the “binomial” in the binomial distribution: two terms. We are not only interested in the number of successes, nor are we only interested in the number of attempts, but in both. Without the other, each is useless to us.

More remedial mathematical symbols:! Factorial: Multiply a positive integer by each smaller positive integer. For example,

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

5!=5 × 4 × 3× 2

Enter the numbers and remember that we have to solve 9 of 10 free throws and 10 of 10 free throws, 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.9^1times.1^0right)

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(9!1!1!×.9.9×.1.1)+(1!1!×.91×.1)

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

= 0.736098929

This is accumulation Distribution, not just possibility distribute. The cumulative distribution is the sum of multiple probability distributions (in our case, this would be two.) The cumulative distribution calculates the chance of hitting a series of values—in this case, 9 or 10 of 10 free throws—not One time value. When we ask Nowitzki what is the probability of 9 in 10, we should understand that we mean “9 in 10 or better”, not “9 in 10”.

If you want to calculate the binomial distribution function for a particular series of events, you don’t have to do it yourself. The helpful people at Stat Trek have a binomial calculator that can do the job for you.All you have to do is provide n, A generation and p value.

So what does this have to do with finance? More than you think. Suppose you are a bank, a lender, and know that the probability of a particular borrower’s default is accurate to three decimal places. How likely is it that the default of so many borrowers will cause the bank to go bankrupt? Once you use the cumulative binomial distribution function to calculate this number, you will have a better understanding of how to price insurance, and how much to eventually loan and keep.

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Have you ever wondered how the initial price of an option is determined? The same thing, kind of.If the volatile underlying stock has p The chance of reaching a certain price, you can see how the stock is in a series n The period determines the price at which the option should be sold.

Applying the binomial distribution function to finance gives some surprising results, if not completely counterintuitive; like a 90% free throw shooter with a 90% chance of hitting a free throw less than 90%. Suppose you have a security that has 20% gain and 20% loss. If the security price falls by 20%, what is the chance that it will rebound to its initial level? Remember, a simple corresponding gain of 20% will not cut it: a stock that drops 20% and then rises 20% will still fall 4%. Maintain a 20% ups and downs alternately, and ultimately the stock will be worthless.

## Bottom line

Analysts who master the binomial distribution have an additional set of quality tools at hand when determining pricing, assessing risk, and avoiding unpleasant results due to insufficient preparation. When you understand the binomial distribution and its often surprising results, you will be far ahead of the public.

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