In america they say "third time's a charm" Three is considered lucky or significant in many religions and cultures eg three lef clovers, the many folk tales and legends that feature 3 tasks, 3 characters etc also many religions feature a trinity besides christianity,
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There's little doubt that the idea is proverbial, as the earlier answer suggests; however, the first time the actual words: "the third time's lucky" appeared in print was in a book of Scottish proverbs published in 1862. You would seem to be right, therefore, in supposing it is not American in origin.
I always thought that you never took the third light from a match - it was thought to have started in WW1 when smoking soldiers would be picked off by snipers, light 1 would alert the sniper someone was there, 2 would show him exactly where so he could "zero in" and three would be curtains - or goodnight Irene! [perhaps apart from the baseball connection is where 3 strikes and you out is from]
A task completed first time would be beginners luck. The 3rd time lucky is an alterative to try try again. Attempted twice & successful on the third time. Just a way of looking in hindsight & it worked out right in the end.