You can see the odds for hatching birds of different rarities in the Hatchery. Just enter the Hatchery, select the type of egg you wish to hatch and tap the "i" button on the left.
The displayed odds are the current average chance for you to get a character of certain rarity. For example, a chance of “1 in 20” translates to a 5% chance for getting a bird of specific rarity. During events with a chance multiplier, as a you open more eggs the chance multiplier increases until you hatch one of the main event prizes. That is also taken into account in the displayed odds.