Going off of the Announcement example, that particular wheel has 9 Spots and a total of 29 prizes. The exact totals may differ depending on what colours you have each time. Pretty sure sometimes I've had 32 prizes on a wheel in the past

It'd be rather difficult to verify if Gold is a 1 in 9 chance, or a 1 in 29 chance. Mainly because the probabilities are constantly changing. So without a official source of confirmation you'd need to not only record every single spin, but you'd need to have a large enough sample size and make sure the math is watertight.

This would likely mean having to sort your results into many different pools of results, running the numbers separately for number of spots remaining and number of prizes remaining. Then comparing and cross examining all the results. However given sometimes you'll go through the entire wheel before you get the Gold and other times you'll get it first spin, it'll take a lot more then just a single city on a single event doing free to play to verify anything. You'd need to be doing it either on every world there is, or be spending a large enough amount of diamonds to be able to record enough spins to compare at each possible possibility

And once you've done all that you still might not have confirmed anything if you end up using the wrong math formula