For an event to occur, in an interval of 28 days, you have a

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For an event to occur, in an interval of 28 days, you have a chance of 0 for the first 23 days and then chances [math] p_1, p_2, p_3, p_4, p_5 [/math] (each in [0,1], no correlation between the p_i's) in the remaining 5 days.

You probe this setup at a random day, the particular day you choose is given by a probability P (with [math] \sum_{n=1}^{28} P(n) = 1 [/math]).

What are the chances for the event to occur?

For starters, I think for P(n)==1/28 constant, you get
[math] \dfrac {1} {28} \sum_{n=i}^{5} p_i [/math],
but that's just my intuition.