Although in formal speech and in textbooks and dictionaries you will find "réussir un examen" but in informal speech, especially "chez les jeunes" you will almost, always find "avoir un examen" e.g.
"J'ai mon bac" - I have passed my baccalauréat.
Hope this helps,
When talking to native speakerd, the distinction between avoir un examen and passer un examen seems not born out in everyday language use. Passer un examen is, apparently, often interpreted as having passed it and not just taken it.
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