The coming of the Holy Spirit wasn’t random timing.
Leviticus 23 (excerpts with celebrations underlined)
The Lord’s Passover begins at twilight on the fourteenth day of the first month. On the fifteenth day of that month the Lord’s Festival of Unleavened Bread begins; for seven days you must eat bread made without yeast . . . Bring to the priest a sheaf of the first grain [First Fruits] you harvest . . . From the day after the Sabbath, the day you brought the sheaf of the wave offering, count off seven full weeks [Feast of Weeks – Pentecost].
Jesus was crucified on the Passover, buried on the Festival of Unleavened Bread and raised from the dead on the Day of First Fruits. The Holy Spirit came on the 50th day, Pentecost.