By chance; accidentally; unexpectedly — indicating something happening without planning or intention.
駅でたまたま大学の友達に会った。
I happened to meet a college friend at the station.
彼女はたまたまその記事を読んで興味を持った。
She happened to read that article and became interested.
たまたま通りかかっただけで、この店に入ったわけではない。
I didn't just happen to pass by and enter this shop (implying a different reason).
Occasionally; once in a while; now and then — used to indicate infrequent occurrences (similar to 'tama ni').
たまたま休日に映画を観に行くことがある。
Occasionally I go to the movies on my day off.
彼はたまたましか顔を出さないので、話す機会が少ない。
He only shows up occasionally, so there are few opportunities to talk to him.