I believe that here it is necessary to evaluate the probability of getting bingo necessarily in relation to the time interval.
(1)
We roughly know when the Moment of Truth will come.
We take an interval, say 6 hours, around this Moment of Truth and count how many of all postcards will be registered at that time.
We count how many of your sent postcards fall into this interval.
If the probability of each of your postcards falling into this interval was 1.0 (100%), then it would be enough to divide the number of your postcards by the total number of all postcards in the interval.
This would be the answer what is the probability of getting bingo in the interval of 6 hours.
But it all is not so easy
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(2)
Therefore, for each of your postcards, we evaluate the probabilities of falling into this interval. How to do this is a topic for a separate discussion. But for simplicity, you can do it rough by eye, choosing for each postcard one of several values āāfor simplicity (0.0 or 0.25 or 0.5 or 0.75 or 1.0). Since the exact calculation is a rather difficult task.
And then sum the quotients of these probabilities for all your cards in this interval.
Letās assume that two of your postcards fall into the interval. The first one falls there with a probability of 1.0 (100%), the other with a probability of 0.5 (50%). And the total number of all postcards falling into the 6 hours interval, letās say 2000 postcards.
Then you need to calculate:
(1.0/2000) + (0.5/2000) = desired chance to get bingo in the 6 hours time interval
And now the most interesting.
Next, you need to estimate the limit to which this value tends when narrowing the interval to zero width relative to the Moment of Truth
And of course, the probabilities of falling into a narrowing interval for each of your postcards will begin to change. They directly depend on its width.
But it is possible to act more simply and evaluate this limit, narrowing the interval only to the width of the interval in which the Moment of Truth walks back and forth and does not reliably go beyond its boundaries.
PS If you look at my graph, the light blue tube is the interval in which the actual value of the number of registered postcards walks with a probability close to 100% and does not go beyond its boundaries. The width of this tube is about 2 hours.
The farther from the Moment of Truth - the wider it is, the closer - the narrower it is.
This tube is called the confidence interval.
And in order to evaluate the probability of getting bingo, it is enough to use an interval with a width in which the fact of the value walks with a probability close to 100%.