odds of doubling up in NL when shorstacked

[TStoneMBD]

game is a NL cash game. you do not have money to rebuy and you will stay until you double up or bust.

average stack at the table is 15xbb. you are the short stack at the table with 4bbs. the second smallest stack is 7bb and there are a 2 players with around 20bbs.

you understand the math of the game well and you play a shorstack fairly optimally.

everyone's skill level at the table is dead average assuming equal chip positioning.

you are on the button.



what are the odds of you doubling up with your size stack before the bb gets to you? what are the odds of seeing the button again if you play your blinds mathematically optimally?

being on the button, what are your odds of doubling up before you see the button again?
what are your odds of doubling up if you start your rotation from the CO? MP3, MP2, MP1, UTG3, UTG2, UTG1, BB, SB?

(i understand that these are all rather small stack sizes for cash games. however, this question is geared towards MTT play. i made it a cash game because i want this question to be answered based on ev, without payout structure having an influence.)