You can add up how many cards it says is in the chest in total (add the 4* + 5* + 6*+ 7* number that is says is remaining in your chest together) to see what the odds are for pulling a particular rarity of card on any pull.
And remember, as you pull from the dream chest your odds get a bit better with every pull UNLIKE the small odds of pulling a 6* each time you pull a premium pack (It is .0-something % chance for a 6* if I remember correctly? And yes that is POINT ZERO something percent, maybe there are even two zeros there I can't remember exactly but it's a very small chance.) That's why most people who have been playing a while only pull packs if trying to rank in an event, so pulling them more for the event items than the cards.
But just like the lottery or a slot machine casino floor, people get randomly lucky and that news spreads pretty fast (especially due to the rare card report, which we all are forced to look at if we want to advertise our cards for trade) and some of us think "Oh, maybe that could be me!"