The chance to get dragonite from the daily dungeon is actually 60%.
30% for opening 3 doors containing 3 battles (3/6 x 2/5 x 1/4 x 3! = 6/20 = 30%).
10% for opening 4 doors containing 3 battles, with the first one being an altar room (1/6 x 3/5 x 2/4 x 1/3 x 3! = 10%).
10% for opening 4 doors containing 3 battles, with the second one being an altar room (3/6 x 1/5 x 2/4 x 1/3 x 3! = 10%).
10% for opening 4 doors containing 3 battles, with the third one being an altar room (3/6 x 2/5 x 1/4 x 1/3 x 3! = 10%).
30% + 10% + 10% + 10% = 60%.
Edit: Me stoopid, it’s really just 10%. The permutation is already included, it shouldn’t be factored in twice. 