Profession Skill Up Chance

I've acquired 50+ of each of my chosen meats for the upcoming push for realm first: illustrious cooking. Do I need more? Do I want more? How safe am I, really?

Those are the questions I was asking today and turned to google to see if anyone actually knew the answer. I couldn't find any links to data from any actual studies but I did find in a few spots the same formula which makes some intuitive sense. If the formula is right then it also explains the weirdness in terms of when the different cooking recipes turned green according to tier.

Orange is always 100% to skill-up (other than skinning which Blizzard intentionally changed a long time ago). Each recipe now has two relevant numbers. The last skill level which is 100% to skill-up and the first skill level with is 0% to skill-up. Every level between those two has a chance of providing a skill-up which is linearly related to those two numbers. The formula is (grey skill - current skill) / (grey skill - yellow skill + 1). The green level is artificial and represents a less than 50% chance to skill-up, and hence lies halfway between the grey skill and the yellow skill. (This is why I can get 10 yellow ticks out of a tier 1 food and only 5 from a tier 3 food. They have a wider range before they go grey.)

Anyone who bothered to read my two part series on Markov chains should see where I'm headed here. Skilling up is strictly increasing and I now have quantified probabilities for advancing from state to state. How much food do I need? I can build a matrix and find out! (Well, I can build 3 matrices and find out since they all have different ranges between yellow and grey.) Each of the three tiers has 15 points in orange, so for each tier I need 15 meat plus 10 skill-ups.

Building up my matrix and running it on the online tool I found shows that for tier 1 I expect to need 12.941 meat to get those final 10 points, for a total of 28 meat. The 64 snake eyes I have should be more than sufficient. In fact, it's a practical guarantee. The odds of not getting 25 skill points with 64 meat is 0. Experimenting a bit, the 95% threshold is 32 pieces of meat.

For tier 2 we're looking at needing 16.5583 extra pieces of meat. The 95% threshold (including the orange 15) is 38. I have 57 blood shrimp which is 99.9988% likely to get me to 500.

For tier 3 we're looking at needing 32.2187 extra pieces of meat. The 95% threshold (again including the orange 15) is 71. I have 59 crocolisk tails which is 85.55% likely to get me to 525.

I'm happy with my practical guarantee to get through tier 1. Missing 1 of every 100000 tries at passing tier 2 is acceptable. But failing 1 in 7 at getting through tier 3 is not something I'm happy with. Fortunately for me I have an easy out. Vek has had enough double token days that I can afford to buy a 4th recipe. I was hoping I wouldn't need to but if I do then I have 114 combined tier 3 meats which would give me a 99.92% to make it. I was thinking I should save the tokens in case I needed to buy a tier 2 recipe but this shows me that's wrong. Even with 2 recipes I'm still more likely to fail at tier 3 than at tier 2. I should be good and I will absolutely make sure to have that bonus skill from a daily sitting around in case disaster strikes. (And beg for more food!)

On the plus side, I doubt other people are thinking they need to stock up hundreds of the tier 3 meat so if I need to make that run other people might have to as well. Better make sure to log out in frost spec for the extra mounted run speed!