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I was thinking about marriage proposals. If the probability of someone saying yes to the proposal is unvarying, say $p$, all one needs to do to get to a yes is to ask enough times. The probability of getting a no n times is $(1-p)^n$, which will surely converge to zero for any $p>0$.

I started to wonder, what if the successive probability of a 'yes' event decreases with each trial? In this case, the probability of getting a no n times is $\prod_{i=1}^{n} (1-p_i)$ and $p_1>p_2...>p_n$. This would perhaps be a better representation of reality.

What can be said about this series as n goes to infinity, or in general? I know the value must be between 0 and 1, but not much more than that.

Is any monotonic decrease in $p_i$ enough to make this value not approach to 0 as n approaches infinity? Or are there some conditions on the 'rate' of decrease in $p_i$ for this value to not be zero?

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  • $\begingroup$ It can certainly be non-zero, for example $\prod_{i=1}^{\infty}(1-\frac{1}{2^i})\approx 0.2887880951$ $\endgroup$
    – Sil
    Feb 22 '21 at 11:38
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    $\begingroup$ The behavior is basically determined by whether $\sum_{i=1}^\infty p_i < \infty$. To see why, take the logarithm of the product and use that $\log(1-x)\sim -x$. $\endgroup$
    – Clement C.
    Feb 22 '21 at 11:39
  • $\begingroup$ en.wikipedia.org/wiki/Infinite_product#Convergence_criteria $\endgroup$
    – user
    Feb 22 '21 at 11:57

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