"I met him by chance"
Let's think of science - you know that many inventions were made unintentionally, by pure chance, right? How do we say it in Hebrew? Like this - הַרְבֵּה הַמְצָאוֹת נַעֲשׂוּ לְגַמְרֵי *בְּמִקְרֶה.
You notice that there is no "by" there, right? Hebrew is less productive in terms of prepositions, pretty much everything turns to "BE-" :)
So the word מִקְרֶה on it's own, when left without BE-, means "chance". בְּמִקְרֶה - by chance. Look at this word - doesn't the root look familiar? Surely you heard people say "?מָה קוֹרֶה" sometimes. We know that it means something like "what's up?" but the literal meaning is "what's happening?"
Like a child in some movie could tell his mom "mom, don't worry, what on earth can happen to me in several hours?" אִמָּא, אַל תִּדְאֲגִי, מָה כְּבָר יָכֹל לִקְרוֹת לִי תּוֹךְ כַּמָּה שָׁעוֹת?!?" And then, well... I leave it to your imagination. :)
Hopefully, no מִקְרֶה חֵרוּם - no emergency case... You see - for "case" we also use this word! It makes total sense - we are talking about something unexpected here!
More examples: (in which often in English we could just go ahead and use "happen"):
יֵשׁ לְךָ מַבְרֵג בַּמִּקְרֶה? Do you (happen to) have a screwdriver, by any chance?
בְּמִקְרֶה אֲנִי יוֹדֵעַ אֶת הַתְּשׁוּבָה!I happen to know the answer!
*And let the ה at the end not confuse you! It's not a feminine word - like in מה הוא עושה -"E" is a masculine ending. :)