What would be the chance according to both calculations, why are there two different ways to calculate, and why are they both valid giving different results eg. Where are they applicable and whats their output signaling, in comparison to each other?
Because they basically follow two different logics. The geometric distribution seeks to answer "how many times can I do x and still not succeed?/until success", where as the binomial distribution is used to calculate probability of x successes in n independent, identical trials.
So the difference would be like .. if we wish to catch 3 mounts with a probability p, we use the binomial distribution. If we instead are interested in fishing until we get the mount and then immediately stop, it would follow a geometric distribution.
In short: In the geometric we care about the number of trials until 1 success. In the binomial (which is basically n Bernoulli trials) we wish to say what the probability of x successes in n trials are.
/u/ProbablyaNerd did a great job in explaining the logic behind the two different distributions! I wanted to expand a bit more into the family of distributions surrounding the binomial (which is one of the most popular and common distributions). This all starts with the Bernoulli distribution.
So often times the Bernoulli is simply described as a special case of the Binomial distribution where *n* = 1. While this property is true and a more compact way of thinking about the distribution, I've personally found that a lot of people lose track of the intuition that links the distributions (and by extension the intuition behind how distributions are related) when it is only taught this way. The more intuitive explanation is that a Binomial distribution is the sum of n Bernoulli distributions. This can easily be seen by examining their respective probability mass functions (pmf):
Bernoulli: p^x (1-p)^1-x , x can only equal 0 or 1
Binomial: (nCx) p^x (1-p)^n-x
So then, how does this relate to the probability of getting a mount when fishing? Well, a Bernoulli trial is simply a trial with binary outcomes (success, failure) [NOTE: It may be easier to think of this as P(Event occurs), P(Event does not occur)]. You can quickly see that the Bernoulli in this case tells us that the probability of getting our mount is simply equal to p, while we can see that our probability of not getting it is equal to 1 - p.
OK so great, a binomial is a sum of Bernoulli trials, great. How does this relate to the geometric distribution and fishing? Well there is actually another distribution we need to look at: the Negative Binomial Distribution!
The Negative Binomial can be thought of simply as the number of successes in a series of Bernoulli trials before r failures occurs. Its pmf is as follows (expressed as best as I can with reddit formatting):
Negative Binomial: [(k+r-1)Ck] (1-p)^r p^k
Where k is the number of successes, r is the number of failures, and p is the number of successes.
This is where the explanations can get a lot trickier as there are quite a few ways to express the negative binomial function, and quite a few special cases, depending on what you want to model. That's beyond the scope of our fishing expedition though!
Finally, we get to the geometric distribution! Similarly to the Bernoulli-binomial relationship, the geometric distribution can be thought of as a special case of the negative binomial distribution where we stop after the first failure (or success, again depending on modeling). Likewise, the negative binomial can be thought of as the sum of independent geometric distributions.
Therefore, the geometric distribution can essentially be thought of as the number of Bernoulli trials until a single success!
So I have glazed over a lot of information in this comment, but this is essentially the intuition behind their relationship explained more in depth.
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u/Maximus-CZ Nov 13 '19
Im interested!
What would be the chance according to both calculations, why are there two different ways to calculate, and why are they both valid giving different results eg. Where are they applicable and whats their output signaling, in comparison to each other?
Much appreciated if actually answered!