On the wiki, there's a section that tells you what any given fish produces:

Fish Pond: Produce

What you might've missed is the percent chance that anything will be produced:

The chance that a Fish Pond will produce an item starts with the equation (0.15 + 0.08 * population of pond).

In other words, as the population of a pond ranges from 1 to 10 fish, the chance of producing anything ranges from 23% to 95%. There's always a chance the fish pond won't produce anything.

I'm not sure how many Sturgeon you have in your pond, but if you have fewer than 10, the chance of getting Sturgeon Roe is significantly lower. For example:

10 Sturgeon -> 95% chance of anything -> 25% chance of 2 Roe; 100% chance of 1 Roe
9 Sturgeon -> 87% chance of anything -> 80% chance of 1 Roe
8 Sturgeon -> 79% chance of anything -> 60% chance of 1 Roe
7 Sturgeon -> 72% chance of anything -> 60% chance of 1 Roe
etc.

In other words, there's a 5% or greater chance that nothing is produced. If anything is produced, then it depends on how many Sturgeon you have. And even though all the checks are done until one succeeds, the chance gets lower at each level, so it's quite possible to end up with no Roe at the end.

The real chance that 9 Sturgeon will produce a Roe, then, is the sum of:

13% nothing
87% * 80% = 69.6% 1 Roe
87% * 20% * 60% = 10.44% 1 Roe
87% * 20% * 40% * 40% = 2.784% 1 Roe
87% * 20% * 40% * 60% * 20% = 0.8352% 1 Roe

69.6 + 10.44 + 2.784 + 0.8352 = 83.6592% chance of 1 Roe with 9 Sturgeon

Similar math can be done for other numbers of Sturgeon.

Now if you only have, say, one Sturgeon, then the math is:

23% chance of anything * 20% chance of 1 Roe = 4.6% chance of 1 Roe

So chances are you'd get 1 Sturgeon Roe in 22 days.

Now of course over time your Sturgeon will grow, as long as you keep fulfilling their requests, so the chance will increase. But the chance is a lot lower than you might think from the "20%" given on the wiki page, due to the initial chance that the Fish Pond will produce nothing at all at any level of fish.

On the wiki, there's a section that tells you what any given fish produces:

Fish Pond: Produce

What you might've missed is the percent chance that anything will be produced:

The chance that a Fish Pond will produce an item starts with the equation (0.15 + 0.08 * population of pond).

In other words, as the population of a pond ranges from 1 to 10 fish, the chance of producing anything ranges from 23% to 95%. There's always a 5% or greater chance the fish pond won't produce anything.

I'm not sure how many Sturgeon you have in your pond, but if you have fewer than 10, the chance of getting Sturgeon Roe is significantly lower. For example:

10 Sturgeon -> 95% chance of anything -> 25% chance of 2 Roe; 100% chance of 1 Roe
9 Sturgeon -> 87% chance of anything -> 80% chance of 1 Roe
8 Sturgeon -> 79% chance of anything -> 60% chance of 1 Roe
7 Sturgeon -> 72% chance of anything -> 60% chance of 1 Roe
etc.

These are just the initial checks. The game proceeds through several levels of checking to produce an item, but the chances get lower at each level. We can sum up all the levels to calculate the full chance that 9 Sturgeon will produce a Sturgeon Roe. (Notice how each level starts with the % that the previous level didn't produce.)

13% nothing
87% * 80% = 69.6% 1 Roe
87% * 20% * 60% = 10.44% 1 Roe
87% * 20% * 40% * 40% = 2.784% 1 Roe
87% * 20% * 40% * 60% * 20% = 0.8352% 1 Roe

69.6 + 10.44 + 2.784 + 0.8352 = 83.6592% chance of 1 Roe with 9 Sturgeon

Similar math can be done for other numbers of Sturgeon.

Now if you only have, say, one Sturgeon, then the math is:

0.23 * 0.2 = 0.046

That is, 23% chance of anything * 20% chance of 1 Roe = 4.6% chance of 1 Roe

So chances are you'd get 1 Sturgeon Roe in 22 days.

Now of course over time your Sturgeon will grow, as long as you keep fulfilling their requests, so the chance will increase. But the chance is a lot lower than you might think from the "20%" given on the wiki page, due to the initial chance that the Fish Pond will produce nothing at all at any level of fish.