Update CSE347_E2.md

pages/CSE347/Exam_reviews/CSE347_E2.md

@@ -114,11 +114,16 @@ $$
 - Assume we run the algorithm $k$ times, and the probability of success is $\frac{1}{2} + \epsilon$.
 - The probability that all trials fail is at most $(1-\epsilon)^k$.
 - The majority vote of $k$ runs is wrong is the same as probability that more than $\frac{k}{2}+1$ trials fail.
-- So, the probability is  
+- So, the probability is 
-  $$\begin{aligned}
+
-  Pr[\text{majority fails}] &=\sum_{i=\frac{k}{2}+1}^{k}\binom{k}{i}(\frac{1}{2}-\epsilon)^i(\frac{1}{2}+\epsilon)^{k-i}\\
+$$
-  &= \binom{k}{\frac{k}{2}+1}(\frac{1}{2}-\epsilon)^{\frac{k}{2}+1}
+\begin{aligned}
-  \end{aligned}$$
+Pr[\text{majority fails}] &=\sum_{i=\frac{k}{2}+1}^{k}\binom{k}{i}(\frac{1}{2}-\epsilon)^i(\frac{1}{2}+\epsilon)^{k-i}\\
+&= \binom{k}{\frac{k}{2}+1}(\frac{1}{2}-\epsilon)^{\frac{k}{2}+1}
+\end{aligned}
+$$
+
+
 - If we want this probability to be at most $p$, we can just solve for $k$ in the inequality make it less than some $\delta$. Then we solve for $k$ in the inequality $\binom{k}{\frac{k}{2}+1}(\frac{1}{2}-\epsilon)^{\frac{k}{2}+1} \leq \delta$.
 
 ## Online Algorithms