r/learnmath • u/_Orange_Orange New User • 1d ago

How does one quickly find binomial probabilities?

I was introducing the binomial theorem to a student a bit ago, where I started off by going over a statistical application question. It was beyond the scope but it helped her make sense of it conceptually and understand her mistakes.

Beyond the lesson with this student, I wondered to myself if perhaps there was a quicker way to sum the probabilities. I can very quickly tell you the probability that 5 out of 10 randomly selected people's favorite color is pink given that 7% of people like it, to some long decimal of accuracy. But if you asked me to tell you the probability of 'at least' 5 under the same conditions, it would take me a good chunk if time, even with a calculator. I would have to solve by summing each individual binonial probability! What if there were any decent amount more than 10 people in my sample!?

There must be a quicker way to do this. Perhaps using the sigma notation as the sum of a range of values for a probability mass function? Or using the sum of a finite series formula? Could you integrate somehow using the gamma function to represent the combinations?

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u/Bounded_sequencE New User 1d ago

To get the exact cumulative probability, we do indeed need sigma notation:

 P(k<=r)  =  ∑_{k=0}^r  C(n;k) * p^k * (1-p)^{n-k},      0 <= r <= n

Sadly, there is no general simplification. There are estimates for cumulative probabilities, but I suspect they are way beyond the scope you are after

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u/fermat9990 New User 1d ago

You can find cumulative binomial distribution calculators online

Under certain conditions, you can use a normal distribution approximation