01. Binomial Distribution

Let's agree on this and prove it with tests - if probability of something is 0.90, your measured experiment can show history of probability as 0.85 or 1.00 for the same "true" result. We know that getting tails while tossing a coin gives 0.50 probability, but if we toss coin only once it will show 1.00 or 0.00 probability

As we keep going it can vary and settle at 0.66, then at 0.53 or 0.49 eventually, over the long time it will settle at around 0.50 but again - it goes to 0.50 only over infinite amount of measurements

Let's plot how exactly amount of measurements correlates with theoretical probability

Probability estimation by coin tosses amount

Use interactive slider to increase amount of tosses

Attempts: 80

Estimation: 0.438Difference: 0.063

As we can see it can take up to 500 coin flips to actually get the theoretical 0.50 probability result. Let's look how different seeds behave over the time:

Probability estimation by coin tosses amount (with RNG seed)

Use interactive slider to increase amount of tosses and button for changing the seed

Attempts: 80

Estimation: 0.475Difference: 0.025

Very interesting, for some seeds it is enough 500 flips to establish at exactly 0.50 probability, for some it takes more time. Let's plot the mean delta from theoretical result over attempts across different seeds:

Mean differences from theoretical probability across many seeds

Use the slider to change the amount of seeds

Seeds: 20