Right now there is an event occurring in Heroes of the Storm where a special hero (Cho'gall) is unlocked if you play with another player currently playing that hero.
I ran into a bit of an intuition problem when I started thinking about the optimal strategy for attempting to unlock this hero.
So you can queue up to enter a match with 10 players total, each of which has an equal probability of playing the new hero (aside from anyone you enter the match with whom you know does not own Cho'gall, like yourself). Myself and one other friend are interested in unlocking this hero, and for all intents and purposes once one of us unlocks it, it is then trivial to share the hero with the friend.
I began thinking that it's more optimal to queue into games separately (as opposed to partying up and getting placed in the same match), because if we take the case where it takes exactly one match with a Cho'gall player to unlock the hero, we essentially get two shots at whatever the probability is that a Cho'gall player shows up in a given match.
However, for some reason when I change the unlock requirement to playing say, 5 games with a random Cho'gall player before unlocking the hero, intuitively I would say queuing separately is no better than queuing together, because if either one of us gets a Cho'gall player in the match, we get only one mark, and it seems like it will take the same average number of attempts to get Cho'gall if we queue separately when compared with queuing together.
If I had to guess, I'd say it has something to do with the Gaussian distribution standard deviation, in that the standard deviation of average number of attempts to unlock Cho'gall under the 5 games requirement is much smaller than the standard deviation under the 1 game requirement, and queuing separately gives us two points on the distribution which is perhaps just less effective with a smaller standard deviation.
Can anyone shed light on my thought process here?
I wrote some code to simulate the process, and the results seem to match some of my intuition in that the more games you need to unlock Cho'gall, the less queuing separately matters.
The code is available here: https://github.com/Riizade/chogall_sim/blob/master/chogall_sim.py
Output
1 game to unlock, queuing separately
average matches: 62.9960000000001
1 game to unlock, queuing together
average matches: 124.7680000000001
5 games to unlock, queuing separately
average matches: 475.6600000000002
5 games to unlock, queuing together
average matches: 636.2030000000009
[Finished in 27.516s]
Even when you need 5 encounters with Cho'gall it would still be better to queue separately. In fact having one person queue and the other not play at all would be better than queueing together! This is because queueing together is like being one player that takes up two slots. That leaves only 8 other players instead of 9. Therefore the chance of encountering Cho'gall would go down.
Of course both of you queueing separetly would reduce the average number of games you need to play compared to just having one person play. Because there is a chance that the second player would unlock Cho'gall first.
However there might be other factors to consider. Perhaps by queueing together you can get faster games because you will have above average teamwork. Also it might just be more fun to play together.