I found this table from the Gearbox website about drops rates. The Badaboom is a legendary item, making its odds of dropping a 1 and 10000 chance.
But that is just for a legendary item. From what I can gather, there are pools of weapons (SMG pool, pistol pool, etc.), manufacturer pools, and more. It also considers what kind of bandit you killed, (regular bandit, badass, super badass, etc.). All of these pools have to be taken into consideration when calculating the odds.
There is also a separate pool that determines what kind of item will drop (such as a shield, grenade mod, weapon, etc.). Here is an image of how that pool looks:
The website describes this pool as:
...a graph showing the relative chances of each item pool being chosen from our raffle. The entire bar represents all of the tickets in our jar. Each colored segment is equal to the number of tickets each pool has and therefore their relative chances of being chosen.
It gets more complicated with things like relics, specifically the Vault Hunter relic, which makes rarer items drops more common.
To sum it up, here's how the game goes through determining what to drop (with a pistol example included):
Without knowing the odds for things like the manufacturer pool and specific weapon pools, it makes calculating the actual odds impossible. I couldn't find these odds on that website. Hopefully I at least gave you an idea of what to expect.