Probability and Variance in game

Shown below is a histogram of a simple coin flip experiment where someone flips a coin 20 times and records what they see each time. Now repeat this a bunch of times.

http://www.mathamazement.com/images/...-histogram.JPG

As you can see, what you might think happens, since the chance of getting heads is 50% we see the highest probability at 10 which is 50% of 20 trials and also the expected value E(X) of the random Variable X.

Variance would be the width of the number of outcomes with probabilities higher than 0, or negligibly 0. In this case between 4 and 16. The wider the range the higher the variance, and vice versa.

Now we can think of it from a crafting perspective by changing the experiment from flipping a coin to Performing Y number of Hasty Touches and recreate a similar type of histogram.

Has anybody tried doing this? From experience crafting in this game it feels like the variance of touch moves and/or HQ'ing something is very wide, which would explain failing a lot of 98% hq, or passing a lot of 6% melds.

[EDIT] This 1000 character limit kills me, you can't have a meaningful discussion with 1000 characters...

You can edit the first post to overcome the limit. The initial post can only be 1000 characters, but then you can edit it and make a book.

Small sample size is what you are seeing. HT success rate seems fine. And we always remember that 98% HQ fail but not the 1% HQ success.

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Originally Posted by dspguy

Small sample size is what you are seeing. HT success rate seems fine. And we always remember that 98% HQ fail but not the 1% HQ success.

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I know, I agree completely. What I was wondering though is if anybody has run any experiments over a much much larger sample size to try and calculate the variance/standard deviation etc..

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Originally Posted by IndigoHawk

You can edit the first post to overcome the limit. The initial post can only be 1000 characters, but then you can edit it and make a book.

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Oooo, did not know that, thanks!

Let Y be a random variable indicating the number of successes out n trials with a probability of a success p.

y | n, p ~ Binomial (n, p).
Expected value = np
Variance = np(1-p)
Standard Deviation = sqrt{np(1-p)}

Example:
We use Hasty Touch (HT) 100 times with the probability of a success 0.5
Expected number of HT successes out of 100 trials = np = 100 x 0.5 = 50
Variance = np(1-p) = 100 x 0.5 (1 - 0.5) = 25
Standard Deviation = sqrt(25) = 5

Plot http://www.wolframalpha.com/input/?i...%3D100+p%3D0.5


If you do not know the probability of a success and want to quantify your uncertainty, then

y | n, p ~ Binomial (n, p)
p ~ Beta(1, 1)
p | n, y ~ Beta(1 + number of successes, 1 + number of trials - number of successes)

Examples:

20 HT, 13 successes.
Beta (1 + 13, 1 + 20 - 13) = Beta (14, 8)
Plot http://www.wolframalpha.com/input/?i...14%2C+beta%3D8

100 HT, 44 successes.
Beta (1 + 44, 1 + 100 - 44) = Beta (45, 57)
Plot http://www.wolframalpha.com/input/?i...5%2C+beta%3D57

Note how much the variances decreases by increasing the sample size.


Further Reading:
http://en.wikipedia.org/wiki/Binomial_distribution
http://en.wikipedia.org/wiki/Beta_distribution
http://en.wikipedia.org/wiki/Checkin...a_coin_is_fair
http://en.wikipedia.org/wiki/Memory_errors

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Originally Posted by Eirny

Let Y be a random variable indicating the number of successes out n trials with a probability of a success p.

y | n, p ~ Binomial (n, p).
Expected val...

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http://assets0.ordienetworks.com/mis...ing%20duck.gif

I don't have any recorded data, but before patch 2.1 went live I crafted over 10K HQ two star food recipes. At the time I was interested in converting my crafting AF into materia, so I didn't use the 100% HQ rotation because of obvious stat limitations when your gear is full of tier I materia for SB purposes. The skill rotation I was using gave me on average an 85% HQ rate (+/- ~% conditions permitting). I can confidently say that more often than not after completing 100 or 200 synths that the actual HQ result was within 1% of the expected result. As for materia melding, I don't even know... all I know is I have some horrible luck.

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Originally Posted by Matsume

As for materia melding, I don't even know... all I know is I have some horrible luck.

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Materia melding is pretty awful... I tend to think that I'm just unlucky but I've had several cases of failing over 10 attempts on a 45% success chance...
I hate materia melding...

The HQ rate feels like it's different from what it says, but it would take a gigantic sample size to get an accurate reading.

For melding, the ~35% success rate on the first meld always seems to be closer to 50% for me; it seems like more than half succeed first attempt, the worst I've had is 6 tries, and at least 90% happen in first 3 tries (should be about 70% chance). The 2nd-4th overmeld have a much worse rate than listed from what I've seen.

I know that Materia Melding can be rough. I failed 35 times on a meld with around 45% a month back. Since FFXI I have always believed that SE has a "Because we say you fail clause." Which always explained to me why you could fail a level 1 craft at lvl 100. I just accept they carried that into this game but that's just a superstition I guess.