#probability mass function

A statistical experiment having successive independent trials having two possible outcomes (such as success and failure; true and false; yes and no; right and wrong etc.) and probability of success is equal for each trial, while this kind of experiment is repeated a fixed number of times (say $n$ times) is called Binomial Experiment, Each trial of this Binomial experiment is known as Bernoulli…

If we flip a coin 100 times and get 59 heads and 41 tails, we may want to estimate the probability it lands heads between 45 and 55% of the time, as well as have a 95% confidence interval.

Using computers, this is simple. We can create a probability mass function of discrete probabilities, ranging from p = 0 heads through to p = 1.0 heads. For each hypothesis, the likelihood of the coin coming up heads = the hypothesis; the likelihood of tails is 1 - hypothesis.

Here I am:

Creating a probability mass function with 1,001 discrete elements, giving us accuracy to the thousandths place.

Updating each hypothesis with 59 heads and 41 tails.

Naively assuming equal prior probabilities for all hypothesis.

For each piece of evidence, after multiplying each prior probability by the likelihood, we can sum up all these probabilities to get a value for p(B) in Bayes theorem: the probability under any hypothesis. We can then divide each hypothesis by this value, normalizing the probabilities to a total probability of 1.

Following these steps with a little bit of python, I find a 95% confidence interval of the coin landing heads between 49.2 and 68.1% of the time. The probability that the coin lands heads between 45 and 55% of the time is only 20.9%.

Realistically, my personal experiences have been quite inconsistent with those of Rosencrantz and Gildenstern (are Dead), so I'd actually assume a much higher prior probability for the hypothesis that the coin lands heads 50% of the time than the extreme values, like 0 or 100% heads.

If we instead imagine a negative exponential function peaking at 0.5, such that p(0.5)/p(X) = (0.5/(0.5-absolutevalue(0.5-X)))^2, we get: 95% confidence interval: 48.6 - 67.1%, and 27.3% chance that the actual value lies between 45 and 55%.

Changing the priors did not have an overwhelming impact, as the large amount of evidence quickly overwhelms it. On the otherhand, if we start with naive priors and then a body of 1,000 coin flips that landed 50% heads to create the priors used for our 100 flips, we get a 95% confidence interval of 47.9 - 53.8%, and a 94.1% chance that it lies between 45 and 55%. The mountain of evidence shaping our priors is not so easily overturned!

  The takeaway here is that, when dealing with quantitative hypothesis, treating them as discrete (eg, 50.5 vs 50.6 vs 50.7% heads) rather than continuous (anywhere between 50.5 and 50.7% heads) makes updating a set of hypothesis a straightforward process: one must simply define a likelihood function. Which, in cases like the coin flip example, is particularly simple.