Praise be to Allah.
In order for an obligatory fast to be valid, it is essential that the intention be formed the night before, because the Prophet (blessings and peace of Allah be upon him) said: “Whoever did not intend to fast before Fajr, there is no fast for him.” Narrated by Abu Dawood (2454), at-Tirmidhi (730), an-Nasaa’i (2332). According to a version narrated by an-Nasaa’i: “Whoever does not form the intention to fast before Fajr, there is no fast for him.” This hadith was classed as saheeh by al-Albaani in Saheeh Abi Daawood. So if someone is still hesitant when dawn breaks, and he is not sure whether he is going to fast or not, his fast is not valid.
It says in Asna’l-Mataalib (1/411): When fasting, it is essential to have a firm and specific intention, as in the case of the prayer, because of the report: “Actions but by intentions…” The intention must be formed before Fajr in the case of an obligatory fast, whether it is in fulfilment of a vow, making up a missed fast (qada’) or an expiatory fast (kafaarah). End quote.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: If a person who is permitted not to fast says on the first night of Ramadan: Maybe I will fast tomorrow and maybe I will not fast, then he decides after dawn has broken to fast, his fast is not valid, because of his hesitation in forming the intention.
End quote from ash-Sharh al-Mumti‘ (6/362)
Based on that, what you must do is make up this day, and you have done well in that regard.
You do not have to offer any expiation, because expiation is only required in the case of a non-traveller who breaks the fast during the day in Ramadan by having intercourse.
See the answer to question no. 49750.
If your sister fasted when she was hesitant, then she must make up the fast, and you should inform her of that.
And Allah knows best.
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