How does Three of Coins work?
Bungie did not give much information about Three of Coins, besides a paragraph in the Bungie Weekly Update on 9/17/15:
Xûr will also sell a new consumable, the Three of Coins, which gives bosses a chance to drop Exotic Engrams, including Weapon Engrams. These consumables have a cumulative luck effect, so even if one doesn’t net you a drop, the next one has a higher chance to do so.
So as we know it, this is how the buff works:
1) You must use a Three of Coins (3oC) to gain the buff.
2) Killing an Ultra (most strike bosses, most raid bosses, Taken champions in patrol, the Ultra Knight on the Dreadnought) removes the buff. On the kill you either get an exotic engram drop, or you don't.
3a) If you got an exotic engram, your increased exotic engram drop rate from 3oC is reset.
3b) If you did not get an exotic engram drop, the next time you pop a 3oC you have a better chance at getting an exotic.
Notes:
As of the recent hotfix, 3oC can't be used as effectively to farm. There is some hidden timer between popping them that affects their effectiveness as of the 9/24/15 Hotfix: "Three of Coins now provides a smaller increase in Exotic Engram drop chance when rapidly killing Ultras"
It is generally observed that the exotic engram drop rate to 3oC use is approximately 1:5, but YMMV. This is purely speculation based on accounts and sampling from friends, the Destiny subreddit, and the Bungie forums.
There are no known statistics, including the "cooldown time" alluded to in the first note, what percentage each coin raises your chance by, or anything like that. Bungie tends to be cryptic about these sorts of things.
It is also possible for most strike bosses to drop an exotic engram without a 3oC. This can be observed in cases of people getting two exotic drops from one boss. This hints, in my opinion, that the 3oC buff is a separate roll that occurs when you take down an ultra.
it used to be that the Three of Coins worked as a method of exotic farming. after the recent patch however, it has been dramatically nerfed. it is designed to be used for the end boos on strikes and story missions, so it now has a hidden cooldown. you now only get the 'chance' of getting anything by using them every 10 minutes, instead of every minute. every time you don't receive anything, the chance that you will get something next time is increased, until you get something. so, basically, just use them at the end for strike bosses and you will get goodies. i recommend using them once a day, therefore you will make them last until next Xur's day