Τυχαία (adverb) means randomly, by chance. So, «έγινε τυχαία» means that something happened by chance.
ex. Συνάντησα τον Γιώργο τυχαία στο σούπερ μάρκετ. = I met George by chance in the supermarket.
τυχαίων: genitive plural of the adjective τυχαίος (masc.) - τυχαία (fem.) - τυχαίο (neut.) (= random)
τυγχάνω: verb
The form we use more in modern greek is the verb τυχαίνω, especially in 3rd person, τυχαίνει = it happens to (do sth) (3rd person in past tense: έτυχε)
ex. Δεν δούλεψα σκληρά γι' αυτό, απλώς έτυχε! = I didn't work hard for it, it just happened (by chance)!
So, in the end, «έγινε τυχαία» = «έτυχε» !
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Vasiliki Baskou, Instructor/Director, https://learn-greek-online.com.