We have a r.v. $X\sim \text{Bin}(30,0.5)$, then which one is true?

Question

We have a binomial random variable $X\sim\text{Bin}(30,0.5)$, then which of the following assertions is true?

• $P(X>15)=0.5$
• $P(X<15)=0.5$
• $P(X>15)>0.5$
• $P(X<15)<0.5$

My approach

I tried to approach this problem by the central limit theorem. Mean and variance are $15$ and $7.5$. After applying a continuity correction, I think the answer should be $P(X>15) = P(X>14.5)>0.5$ but the manual gives the 4th option as the correct answer.

Is my approach incorrect?

Please help me out with this problem.

Thanks.

Answer 1

By symmetry,$$ P(X < 15) = \frac{1}{2} (P(X < 15) + P(X > 15)) = \frac{1}{2} (1 - P(X = 15)) < \frac{1}{2}. $$

Answer 2

Since $n=30$ is large and $p=0.5$ is far from $0$ and $1$, then $X \sim N(np,npq)=N(15,7.5)$. Using continuity correction factor: $$P(X>15)=\Phi\left(Z>\frac{(15+0.5)-15}{\sqrt{7.5}}\right)=\Phi(Z>0.18)<0.5 \\ \ \text{or simpler:} \\ P(X>15)=P(X\ge 16)<0.5.$$ Now note: $$P(X<15)=P(X>15)<0.5.$$