Depends.
Over the next 2 years, I expect SpaceX to send a lot of probes out.
First, mass to orbit is super cheap now. Second, part of the whole R&D for Mars will be to try a variety of long range engine types and configs, and they love to iterate fast and play.
So I expect probes sent in Mars' direction, but also elsewhere to explore. And I cannot stress this enough, they are clearly capable engineers, but these probes don't need 10 years of development and hyper engineering, because cost to orbit is meaningless.
A probe fails? Who cares! You sent 5 different designs, and can just send more anyhow.
NASA was concerned about cost to orbit, but also perception. A lost probe would be a "waste to taxpayers", and just getting launch windows was a chore. I remember some probes had to wait years for space on the shuttle, so constrained was lift frequency.
But SpaceX is OK with blowing stuff up during tests, with learning as they go. So I imagine 2 years max, they'll be spewing satellites all over the place, without a concern if they fail early, or even make it.
Yet some will. Many will.
And you can bet some will head to asteroids.
Note: at this point someone usually chimes in with how hostile space is, and cheap probes won't work, and yada yada.
I didn't say cheap. Cheap implies skimping out on part quality or construction. You don't need to do that, as the parts are not remotely expensive compared to R&D, dev time and custom build work. In fact if one removes R&D, part cost isn't noticeable.
This means that scale is cheap. Building 1000 probes is almost the same cost as 1. And launch is now super cheap.
Access to Space is on the verge of when airplanes became unremarkable.
And it's coming faster than we think.