〜まい
will not (resolve); probably not
Verb dictionary form + まい (する → するまい / すまい)
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まい is an old negative volitional, and it has two uses.
Resolve: the speaker's determination not to do something: 二度とあの店には行くまい, "I'll never go to that shop again". It's often followed by と決めた or と思う.
Negative guess: "probably not", like ないだろう: こんなにいい天気なら、雨は降るまい, "with weather this good, it probably won't rain".
It follows the dictionary form of most verbs. With する and 来る, several forms exist (するまい, すまい, 来るまい, こまい).
It's formal and literary. In conversation, ないつもりだ and ないだろう are far more common, but まい appears in novels, speeches and set phrases.
Examples
二度とあの店には行くまい。
I'll never go to that shop again.
もう彼には頼むまいと決めた。
I've decided I'll never ask him for anything again.
こんなにいい天気なら、雨は降るまい。
With weather this good, it probably won't rain.
彼がそんなことを言うまい。
He surely wouldn't say such a thing.
失敗を繰り返すまいと、毎日練習した。
Determined not to repeat my mistake, I practised every day.