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I think the Golden Pig machine should be considered bugged. The chance for getting any legendary item if your rainbow bar is not full appears to be vanishingly small. How many different weapons and off-hand items does it give? I think there are at least 8 weapons and 2 off-hand weapons for each bird. If so, the number of non-legendary items is 2 x 8 x 5 = 80. Since for each you get a chance of 0, 1, 2, or 3 stars. there is a total of at least 320 different non-legendary items you can get. For legendary items there are 2 items/set x 3 sets/bird x 5 birds = 30 legendary items.
So if each possibility has an equal chance, your chance of getting a legendary item is less than or equal to 30/320 = 0.094.
If anyone has any additional data (more non-legendary items awarded by golden pig, real statistics on legendary items), please provide.