Suppose $E(2^X)=4$. Prove that $P(X \ge 3) \le {1 \over 2}$. Suppose $E(2^X)=4$. Prove that $P(X \ge 3) \le {1 \over 2}$.
I thought that this question could be solved by using Jensen's inequality and Markov's inequality like below...
$E(2^X)=4 \ge 2^{E(X)}$ which implying
$2 \ge E(X)$
$P(X \ge 3) \le {E(X) \over 3} \le {2 \over 3}$
How I can get  $P(X \ge 3) \le {1 \over 2}$ from this??
Thank you.

There's no need for such powerful tools.  If $X \ge 3$, then how large must $2^X$ be?
