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If one starts praying at a time that he/she knows that it is already prayers time having relied on the islamic calender, and during the course of the namaz, the call of adhaan for that particular prayer is heard. Is that persons prayers valid or void? Should they pray again for precaution?

In the name of 'Allah, - Al-'Aliim.

The Muadhdhin [Caller to Salaat] also depends on the time to which prayers are offered. Hence, It is a matter of knowing the time and not hearing the Adhaan. Although, it is better... rather an act of great reward to wait for the Adhaan, listen to it, follow it and recite the Iqaamah after it. As per a hadith, one who recites the adhaan followed by iqaamah, hundreds, if not thousands, angels in lines pray behind him/ her reaching to the heaven.

As per your question, if you've followed the timings for prayers accordingly, then you're not required to pray again because you've neither hastened nor delayed [Kazaa]. 'Allahu a'lam.

The elders can also share their views, insha'Allah.

Salaamun 'aleykum.