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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!

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  • $\begingroup$ Binomial series are among the most applied and useful mathematical concepts. But also it is important to know that "math for the sake of math" is not a negative thing. $\endgroup$
    – Trebor
    Commented Apr 6, 2023 at 7:00
  • $\begingroup$ Oh, I completely agree! That's a lot of math, of course, but with some concepts I like to find a tangible use to wrap my head around it a bit more. $\endgroup$
    – StayGoldPaulyBoy
    Commented Apr 20, 2023 at 23:11

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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.

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  • $\begingroup$ (+1) I use the binomial distribution a disturbing amount when it comes to playing and analyzing video games (but that's to be expected with so many repeatable success-or-fail scenarios). Be it hunting for Shiny Pokemon, or drop rates in Final Fantasy, I've used the binomial theorem to death. $\endgroup$
    – PrincessEev
    Commented Apr 5, 2023 at 0:02
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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
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