needing help with min-max inclusion-exclusion

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Problem (from Luogu P3175): Initially, you have the number $$$0$$$. Every second, you randomly select a number from the range $$$[0, 2^n - 1]$$$ and perform a bitwise OR operation (| in C++/C, or in Pascal) with the number in your hand. The probability of selecting number $$$i$$$ is $$$p_i$$$. It is guaranteed that $$$0 \leq p_i \leq 1$$$ and $$$\sum p_i = 1$$$. Find the expected number of seconds until the number in your hand becomes $$$2^n - 1$$$.

Two ways to calculate the expected minimum time of choosing one element in set $$$S$$$:

$$$ E(S)=\dfrac 1{1-\sum_{T\subseteq\complement_U^S}a_T} $$$ $$$ c_i:=\sum_{j\ \mathrm{bitwise\_and}\ 2^i=1}a_j\\ E(S)=\dfrac 1{1-\prod_{i\in S}(1-c_i)} $$$

The first one is right, and the second one is wrong. When we have $$$n=2,p=[0.902611887323,0.012000014846,0.032707823145,0.052680274686$$$, the correct answer is about $$$16.9037009548$$$, but the second algo will give $$$20.253655892945$$$. The first one can get AC in P3175, but the second can't. Why?

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