Binomial distributions | Probabilities of probabilities, part 1

What people said

Recommends Varad Mahashabde · 2020-04 · Mathematics Stack Exchange · on “Recalibration of probability estimate when considering multiple future outcomes”

@Tavish Because $\frac{8}{20}$ is the most likely probability of $P(T)$, doesn't mean that $P(X_{next} = T)$ is the same. Other probabilities can also give this distribution even if it is the most likely for $\frac{8}{20}$. Accounting for those gives the formula-esque thing shown in the question. 3blue1brown has an (ongoing series)[ [this video] ] about deriving on YouTube. (1/2)

Recommends Wolf Elkan · 2022-03 · Mathematics Stack Exchange · on “I need help with math statistics for a game.”

There is a surprisingly simple solution to this, it's called Laplace's Rule of Succession . What you do is pretend each player played two extra "fake" games. Of these, they won 1 and lost 1. Then, calculate their percentage based on these games as well. The more games each player plays, the less their score will be affected by the fake games. For example, your player with $250$ wins and $7$ losses, instead of calculating their win rate as $\frac{250}{257}\approx97.276\%$ , pretend they actually had $251$ wins and $8$ losses. This would give them a WR of $\frac{251}{259}\approx96.912\%$ . Your player with $3000$ wins and $400$ losses would have a WR of not $\frac{3000}{3400}\approx88.235\%$ b

Recommends jskherman · 2023-06 · Mathematics Stack Exchange · on “Correct application of Laplace's Rule of Succession to actual effectiveness in Student Problem Sets?”

I recently came across 3blue1brown's YouTube short [LINK] on Laplace's Rule of Succession , as well as his two videos on the probability of probabilities [this video] . In these videos, he uses the example of Amazon reviews to explain how Laplace's Rule of Succession can provide a better estimate of the likelihood of having a positive experience with a specific seller. The rule suggests adding one positive and one negative review to the available data and then calculating the overall rating. I'm interested to know whether Laplace's Rule of Succession can also be applied to evaluate a student's performance in solving problem sets . Specifically, I'm curious if adding one positive and one negative

Cites Danyu Bosa · 2020-10 · Mathematics Stack Exchange · on “Binomial distribution with uniform parameters”

Researchers have found a COVID vaccine that causes harmful effects with probability $p$ where $p$ is uniformly distributed in the interval $[0, 0.5]$ . To check the effectiveness of the vaccine, the researchers test the vaccine on $10,000$ volunteers and find that no one experiences adverse side effects. What is the smallest real number $\lambda$ such that the researchers can assert $p < \lambda$ with probability at least $90\%$ ? I just thought of this problem after I read about how COVID trials have started. I drew inspiration from the 3blue1brown video [this video] on binomial distributions. Unfortunately, that is all the context I have, so how I would I go about solving this problem with thos