I realise there is probably no way to get an accurate figure, just looking for a ballpark estimate of approximately how many board games there are out there, and how many of those involve dice rolling.
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BoardGameGeek has 58627 games listed*, and 11132 of them are marked with the dice-rolling mechanic** . So that's 19% - but of course, there's no guarantee that all of the games were appropriately tagged, so treat it as a lower bound.
If you're curious about more specific categories, you can used the advanced search and limit it to games with or without dice rolling with some other restrictions. But it maxes out at 1000 results (ten pages), and doesn't count them for you.
* Hover over "browse" on the home page