Single-event vs multiple-event probability?

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I may be asking something basic, but would nevertheless appreciate the answer.

Say, for example, we have 10 trials of the event with a single-event probability of success that equals 0.5. I want to find out what is the likelihood of having at least 5 successes.

The binominal calculator says it's ~0.623. However, shouldn't you expect there to be a 100% likelihood of at least 5 successes out of 10 trials in the long run? I mean, if you do an infinite number of such 10-trial sets, it would essentially be the same as doing an infinite number of single trials, right?

Please help me.