The proceeds of marijuana, licensing fees and penalties go into the Washington State dedicated marijuana account (RCW 69.50.535). There is an appropriations rider on this account that stipulates where some of these funds are to be dispersed (RCW 69.50.540).
Following the money from cannabis in Washington State can be tricky. Some appropriations are expressed in percentages and some are harder numbers. Some are expressed in minimums and some in maximums. The answer is really that the appropriations set guidance and minimums that is then formalized by the budget in the legislative process. Additionally, this depends on the amount of revenue that is actually realized.
The expected percentages that are to go to specified programs are to be roughly:
49% Basic Health Care Trust Fund Account
8% Department of Social and Health Services
5% Health Care Authority
4% Local Governments
3% Marijuana education
2.3% Liquor and Cannabis Board
.5% UW research on health effects
.15% State healthy use survey
.05% Washington State Institute for Public Policy study
.01% UW Public web-based education materials