I play a game called Summoner's War. It is RNG-based and as such, has a chance of success for each sample.
Data has shown that in a particular part of the game where one may 'summon' from a small pool of monsters, there is a success rate of 5.67% per sample for a certain type of monster.
Based upon this data, the Expected Value of obtaining this type of monster is believed to be one every ~17.64 samples (100/5.67).
We can now assume that after 17.64 samples that we have a 50% chance to obtain said monster because we know 17.64 is the average number of samples it takes to obtain said monster.
Since the success rate is 5.67% and one has a 50% chance to obtain said monster at 17.64, shouldn't we be able to also calculate the number of samples (17.64) from a binomial formula that equals 0.5, the average?
To reiterate, the formula 1-(1-P)^x = 0.5. Shouldn't x result in 17.64? A 50% chance to summon the monster.
It doesn't seem to work for me.
1-(1-0.0567)^(17.64) results in ~0.64 or 64% chance to summon the monster.
1-(1-0.0567)^x = 0.5 results in x = ~11.8749 or about 11.8749 samples to summon the monster.
What am I missing? Is the average value not 50%, but instead 64%?