Calculating Cumulative Binomial Probabilities

Question

I was given the question to find $P(10\leq X\leq 12)$ with $n=15$ and $p=0.666$. Does this mean that $P(10\leq X\leq 12)=P(X\leq 12)-P(X\leq 10)$? If so, I got the answer $0.9206-0.5959=0.3247$ yet the textbook answer is $0.5390$. What am I doing wrong?

Answer 1

Note that$$P(10\le X\le12)=P(X=10)+P(X=11)+P(X=12)$$

So, $$P(10\le X\le12)=P(X\le12)-P(X<10)$$

Answer 2

No, it is not right. What you have calculated is $P(10<X\leq12)$ since you subtracted $P(X\leq10)$. Correct is to subtract $P(X<10)=P(X\leq9)$. Then, $$P(10\leq X\leq12)=P(X\leq12)-P(X<10) $$