$\begingroup$If I have a certain event that is $100 \%$ to happen exactly once in $10$ days, how can I calculate the probability each day?
I cannot be $10 \%$ per day, since by the end of $10$ days, there still would be a $0.9^{10}$ chance of the event not happening. It can't be $100 \%$ per day either, since that would mean the event happens $10$ times.
What if for smaller and smaller intervals? What's the probability of the event happening in $10$ minutes, $1$ second? $1$ nanosecond?
$\endgroup$11 Answer
$\begingroup$The probability each day is 10%, the probability of the event happening on the 10th day is still 10%, the probability that you calculated is a conditional probability. If we call $A_i$={the event happen the i day}.
{the event happen the 10th day knowing that it had not happen the 9 day before} = $\{A_{10}|\cap_{i=1}^{i=9} A_i^\complement \}$ We can calculate $$ P(A_{10}|\cap_{i=1}^{i=9} A_i^\complement )=\frac{P(A_{10}\cap (\cap A_i^\complement))}{P(\cap A_i^\complement)}=1 $$
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