Probability of losing more than 50 in flip coin

Question

You flip symetric coin 1000 times. For head you get 1\$ and for tail you lose 1\$. Estimate propability of losing more than 50\$.

My attempt:

$X-(1000-X)$ - is random variable of how much we win

$P(2X-1000<-50)=P(X<475)=1-P(X\ge475)$

Now i used markov inequality but it gives negative value so it doesn't make sens.

Answer 1

Markov's inequality predicts $\displaystyle \frac{500}{475}$ which is horrible. Talk about upper bound!

I'd recommend a normal/binomial approximation?

You have $E(X)=500$ and standard deviation $\sqrt{1000(\frac{1}{2})(\frac{1}{2})}$, which means that $P(X<475)$ corresponds to a $z$ score below $-1.58$ and a probability of $0.057$.