probability of coin landing on tails less than 230 times out of 500, and more than 270 out of 500?

Question

I've been stuck on figuring out how to do this with binomial distribution. I know i have to use it, but not how. Is there someone who can help?

Answer 1

Assuming that a fair coin is used and trials are independent, $$X\sim B(500,0.5)$$ where $X$ represents the number of tails thrown out of $500$ throws. For the probabilities desired $$P(X\lt230)=\sum_{k=0}^{229}P(X=k)=0.03330483$$ $$P(X\gt270)=\sum_{k=271}^{500}P(X=k)=0.03330483$$ They are equal in probability because for any distribution $Y\sim B(n,0.5)$ we have that $$P(Y\le k)=P(Y\ge n-k)$$