Statistical Analysis, Statistical Independence

[azure minnow]

What you are describing is independant events versus consequential events

We know a fair coin will always be 50/50, so the odds of any individual flip being heads or tails is exactly 50/50

This is different from the odds of any specific path of events happening. Lets draw a diagram that will branch each time we flip the coin to detail heads or tails (give me a moment)

Using this random image I found on google, we can very clearly see that each flip is still 50/50, but there are an exponential amount of outcomes

The key idea is that each path has an equal chance of occurring

Its intuitive that the path Heads/Heads/Heads/Heads is going to be unlikely because we know that its a 1/2^4 chance of that happening

we need to guess 4 times and be correct

Same goes for tails/tails/tails/tails

But remember that there is nothing inherently special about h/h/h/h or t/t/t/t, other than as humans and thinkers we note that they are all the same value

It turns out that every path is just as unlikely whether its the path of entirely heads or entirely tails; we just associate significantly less value with the ones inbetween since we dont see an obvious pattern

If I wanted to roll specifically a h/t/h/h then, statistically, it would take me the same amount of time on average than if I wanted to roll a t/t/t/t or h/h/h/h

To address your claim a little bit about the 75% odds

Imagine the scenario in which we flip the coins twice: there are four outcomes

HH, HT, TH, TT

You say we are betting on H but the coin comes up T. This means that our first flip is a decided T, and the remaining paths given that are TH and TT

At this point in time, the odds of us winning are still 50%, but the odds of us losing twice in a row are 25%