まいWon't, Intend not to, Probably not

まい and だろう are both used to express speculation, but they contrast in polarity. だろう indicates positive speculation or likelihood, as in 雨が降るだろう (It will probably rain), while まい expresses strong negative speculation or denial, as in 雨は降るまい (It probably won’t rain).

まい and the volitional form (〜よう) both express the speaker’s intent or decision, but in opposite directions. 〜よう indicates intention or willingness to do something, as in 行こう (Let’s go), while まい expresses a strong negative intention or refusal, as in 行くまい (I will not go).3

Though these two phrases appear similar in form, both suggesting 'without doing', they differ in that ともなく・ともなしに is used figuratively to express that the action was performed, but in an absentminded manner, or ‘without even so much as the intention of doing (A), (B)’ e.g. 見るともなく見ていたら 'as I was absentmindedly looking...', whereas ないで literally means that action was not performed, e.g. 見ずにやる 'do without looking'.

まい was previously introduced with verbs as a condensed (and somewhat more formal) way of saying ないだろう. This is that same まい used with the negative form じゃない following nouns, and combined with し to list reasons, resulting in a phrase meaning 'You're/It's not (A), so...' , e.g. 子どもじゃあるまいし ('You're not a child, so...').