The Thlot Plickens... highcough!
The first thing is to think directly on the profit rather than the original investment of the 60gp. That original investment is a given. Even if you get this back by incurring no losses, to sustain this investment in the tavern over time, you will need to keep reinvesting the 60gp each time. It becomes a rolling re-investment. So, for now forget about this.
The aim is to get a maximum profit every tenday, which is 3d10x5gp, or 82.5gp on average.
The average roll of a d100 is 50.5, just like any dice roll, to work out the average you add 0.5 to half of the maximum value of the roll, e.g. for 1d8 the average is 4.5, for 1d4 it's 2.5, etc.
The average roll of 1d100 + 10 is 60.5 (50.5 + 10)
To get maximum profit you need to roll 91 or higher.
So, 91 - 60.5 = 30.5! But, you cannot roll .5 on a die.
You need to go either higher or lower; you can either invest 30gp, or 31gp on marketing.
Investing 30gp on marketing means that you will mostly break even or make a profit, but you will lose your investment on 1 occasion every 100 throws on average.
Investing 31gp on marketing means that you will always come even and not lose your investment, but you will be marginally over-investing.
The profit for investing 30gp is indeed marginally greater than that of 31gp.
However, you can off-set this by alternating your marketing investments; this will give you the maximum profit because in effect your average investment will be 30.5 gp. You will lose your investment every 200 throws on average, yet you will make the highest profit.
Invest 30gp one tenday; invest 31gp the next tenday; and keep alternating!
To sum up:
Have a 60gp as on-going investment; and keep 60gp in the kitty for that odd occasion where you roll a 1 on a tenday in which you had invested 30gp.
Keep alternating your investment in marketing between 30gp and 31gp every tenday. Your average will produce maximum profits and you will get on average 82.5gp back for your regular investment.
It is definitely worth investing in the Tavern!
Note: special thanks to @pwi for pointing some helpful things out and helping me re-visit this problem.