Median values in probability distribution where two median values can possibly occur?

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I have a relatively basic question, I will keep it short, sharp and shiny.

In a probability distribution lets suppose $P_1 =0.5, P_2 = 0.0, P_3 =0.5$

All three outcomes share a slice of the median constant of $0.5$. Which outcome is the median awarded to? I assume the convention is to award it to the first outcome that encounters the $0.5$ value - $P_1$, but I could be wrong, and this is bugging me...

I apologise for the elementary question in retrospect but would love an answer.