If I'm reading the size table right, it's 6% chance for a tiny crown and 3% for a large crown. You can see the distribution table for these investigations in the first column of this spreadsheet (gold crown sizes are highlighted in yellow). These have a decent chance of giving you a crown. HR investigations with at least 1 gold and 1 silver, 3 gold and 1 bronze, and tempered investigations with 3 purple boxes The only exception to this seems to be investigations with 3 gold and 1 bronze boxes, which can give you crowns. An investigation with 2 gold and 2 bronze boxes might seem better than one with 1 gold, 1 silver, and 1 bronze, but it will still not give you crowns (possibly because these usually come with 8 attempts). HR investigations without a gold and a silver box or tempered investigations with only 2 purple boxes cannot give you any gold crowns. Optionals have random monster sizes and HR optionals will give you up to the largest silver crown and the smallest possible size before a tiny crown but will never give you a gold crown of any kind. Assignments/special assignments will always give you the default monster size, except for the Lunastra special assignment, which has a large crown Teostra, and the Kula-Ya-Ku special assignment, which gives you a ridiculously large gold crown. The size table is chosen solely based on the investigation rewards (number of purple, gold, and silver) and has nothing to do with the difficulty or any other investigation settings.ДЄll possible quest types can be broken into a few groups when it comes to the chances of getting crowns:ДЄssignment quests, optionals, LR investigations, and certain HR/tempered investigations

Investigations have a hidden size modifier, but that only seems to increase the chances of getting larger/smaller monsters within its own size distribution table by an unknown amount. I made a post earlier today talking about how the crown sizes are calculated and after spending some time playing around with Cheat Engine and the save editor, I was able to figure out what kind of monster sizes you can get from each quest.

Monster hunter world crowns