Can we truly random choose which finger to raise by counting to 10 in your mind?
I think your question ultimately is about whether people can or are able to behave like a uniform random number generator drawing numbers from 1 to 10.
I don't think the fact that the participant needs to raise a finger changes things much.

Note that there is a distinction between what people do, and what people are capable of. A lot of research suggests that people are not very good at simulating randomness. That said, one can think of strategies that would allow a person to simulate randomness more effectively (e.g., drawing on sources of random information in the environment, or even applying a formula similar to that used to sequentially generate numbers using random number generators).

The broad question has already been posed on the site: 
How well can a human-generated &quot;random number&quot; be predicted?

I also note that a participant could just report sequentially 1,2,3..., 10, 1, ... This data would show 100 of each number for 1000 signals. Of course, this would exhibit structure that would not be present were each trial independent.