O Flappy Bird, The Pipes, The Pipes Are Calling

OK. So I love this game, Flappy Bird.
It's insanely stupid.
But my friends and I are playing it and competing over our best scores.

A couple of us started running stats on the game (of course), and I noticed some trends.

First of all, that my expected high score was much lower than my talent suggests! I get extremely nervous as I get nearer to high scores (among my friends' is currently 61) and end up dying shortly beforehand. So I wanted to quantify this.

First, the most basic principle for Flappy Birds - math (yes, you are still reading this...) is the following:

Where p represents the probability of dying on any given score.

For example: if my average score is 2, I am making it through 66% of all the pipes (2 successfully, failing on the 3rd. 2/3 = 66%). This is fairly straightforward.

Next I wanted to figure out my expected # of rounds by which I successfully get to each score. Couple of steps are required.

1) As of this writing, my (p) value is 94.0527% - (my average score is 15.8.) So I made a chart of probabilities. Probability of getting at least 0 points? 100%. Probability of getting at least 1 point? 94% of course. Probability of getting at least 2 means I've made it through the 94% twice, so 0.94^2 or 88.4%. And so on and so forth. So we now know that the inherent probability of making it to a certain number of points in the game is p^number of points. 

2) By multiplying this probability by the number of trials I have completed, we have determined the number of expected times I should have reached at least __ number of points. I then compare this with the actual number of times I reached that number of points. The resultant chart is what I like to call my FBNG or "Flappy Bird Nervousness Graph":

As you can see I tend to do much better than expected in rounds 10-25, after which I taper off heavily to average/below average # of games finished with said score. So one could justifiably argue that I do fine until I hit 25. I honestly do start nervously looking at the score whereupon it gets into larger figures.

So from this graph I can see how I'm doing versus expected. But I want to know something simpler: what should my high score be? Currently mine is 57, but according to my 94% pipe clearance I can tell that I should be able to do better. In fact, my little chart of numbers shows that I should have at least 1* game of all the way up to around 74 according to the math! But I don't want to use a chart to guess and check. I want a formula.

So I can see that around (p of 0.94) ^ (74 games) is roughly 1. I want to see at what game # my (p) should hit EXACTLY 1.

I derived it like so:
(p^exp high score) * #GamesPlayed = 1
1/GamesPlayed = p^score
therefore
log with base (p) of 1/GamesPlayed = expected high score
or

When you plug this in, you'll see that 74 is close. Plugging in 94.0527% and 97 games gives you 74.61.
My high score is 57, so I've got to get over some nervousness so I can achieve my true high score.



* I thought about using 0.5, but 1.5 is just as relevant, so I settled on 1 which is in the middle.