Bladee's Mean

It's time for the Bladee concert once again! Bladee performs $n$ songs at his concert. To consider the concert a success, he must successfully sing $n$ of his songs, say $1,2,\dots, n$ . The probability that he fails to sing song $i$ is $p_i$ , independent between songs. To calculate the probability $P$ that his concert is a failure, the probabilities $p_i$ are all approximated by the arithmetic mean $\overline{p} = \frac{1}{n}\sum_{i=1}^n p_i$ Let $\overline{P}$ be the approximation for the probability the concert is a failure $(P)$ using $\overline{p}$ . How does $\overline{P}$ compare to $P$ ?

The relationship cannot be determined because AM-GM does not apply to probabilities.

$\overline{P} \le P$ , with equality only when all $p_i$ are equal.

$\overline{P} \ge P$ , with equality only when all $p_i$ are equal.

$\overline{P}=P$ for all choices of $p_i$ .