What is an actual application problem (probability, weather) that uses the binomial series? Does it solve anything?

Question

I'm just trying to figure out what the purpose is of the binomial series? What does it tell us? I did a search and found something talking about probability and weather predicting, but I'd like to see the formula in action. My searches aren't yielding anything yet. If it doesn't actually do that, what does it do?

Is it just a math problem for the sake of being a math problem? (For the record, I'm pretty new to series/sequences in Calc 2, and I have a project by the end of the month about binomial series, but it just helps me to understand tangible uses of math).

Thank you!

Answer 1

Your best bet would be to learn more about the binomial distribution. Basically the mechanics of the formula $$(x+y)^n=\sum_{k=0}^n\left(\begin{array}{l} n \\ k \end{array}\right) x^{n-k} y^k$$ can generate a probability distribution when $x$ and $y$ sum to 1. Since $1^n$ is $1$, the right hand side becomes $1$ and each component of the summation represent the weight of different probabilistic outcomes.

When $x$ and $y$ are both $\frac{1}{2}$ for instance, each component $\left(\begin{array}{l} n \\ k \end{array}\right) x^{n-k} y^k$ represents the probability that a coin flips as heads $k$ times given $n$ coin flips.

Answer 2

Here are some relatively hidden uses of Binomial theorem

1. Binomials are often used on forecasting algorithms
2. For generating and distributing IP addresses of electronic device
3. In the derivation of the most famous equation, Einstein’s $E=mc^2$
4. Models to analyze the impact of policies on the economy
5. Ranking