I did my math a little differently and got slightly different numbers for 4-7 stars

You must get 3 successes in a row

SO from 4 to 5 you have 55 ways to fail with a 45% success rate
you have 45 ways to succeed from 4 to 5 then 68 ways to fail from 5 to 6. Giving you 45 * 68 or 3060 ways to have this happen
45 ways to go from 4 to 5, 32 ways to succeed from 5 to 6 and 80 ways to fail from 6 to 7 so that gives you 115200 possibilities
45 ways to go form 4 to 5, 32 ways to go from 5 to 6, and 20 ways to go from 6 to 7 for another 28800 possibilities

This gives us a total of 147115 possible out comes
28800 give you a 7 star hero
28800/147115 =19.576% chance of success to get a 7 star hero

Now that is assuming you go from 4-7 with out a fail you have 19.576% chance of success.