You don't know if you're an independent thinker

To be more independent-minded, you've to not know what most others think.

Because if you don't know what most others think, you cannot blindly follow their beliefs and hence you will have to be independent-minded.  

But a conventional minded person will probably feel anxious about not...

 Fastidiousness about Truth means being careful about the degree to which you believe in something  

Conventional thinkers often think if something is probable, it is certain.

And if something is unlikely, it's impossible.

But Independent thinkers don't believe anything, about a...

Saying I don't know is an admission of power: You'll be wrong less often, and it will lead to more considerable improvement.

If saying "I don't know" is too difficult, rephrase it terms of intellectual humility. "I don't know how to program" becomes "Learning to program is...