If $X$ follows a binomial distribution, then what is $P(1.25<X<1.75)$?

Question

Do I understand it correctly that $$P(1.25 < X < 1.75) = P (X <1.75) - P(X < 1.25)= P(X \le 1) -P(X \le 1)=0?$$

Another question : Is $$P(0.5 < X <1.5) = P(X =1)?$$

Thanks in advance.

Answer 1

Yes. If $X \sim \operatorname{Binomial}(n,p)$, then $\Pr[1.25 < X < 1.75] = 0$, because the support of $X$ is on the set $X \in \{0, 1, \ldots, n\}$. Since there are no integers between $1.25$ and $1.75$, the desired probability is zero.

Answer 2

Another way to look at it is that the event $1.25<X<1.75$ is an empty event since the only possible values of $X$ are integers $k$, $0\leq k\leq n$. Hence $$P(1.25<X<1.75) = P(\varnothing) = 0.$$

For the second part, notice $$\{.5<X<1.5\}\iff \{X = 1\}$$ and hence $$P(.5<X<1.5) = P(X=1).$$