Solve $P(2\leq X\leq 4) = P(3\leq Y\leq 6)$ using mental arithmetic only

Question

Question: Suppose that $X\sim B(4,p)$ and $Y\sim B(6,p)$ where both $X$ and $Y$ are binomial random variables.

What is the $p$ that satisfies $$P(2\leq X\leq 4) = P(3\leq Y\leq 6)?$$

I understand that $p$ can be found by solving the equation above and the probability mass function for binomial distribution.

However, I am trying to solve it in a 'smart' way, that is, without using pen and paper, solely using mental arithmetic only.

Is it possible to achieve it?

Answer 1

Let $Z$ be a $B(2,p)$ independent from $X$. Denote $A$ the event $X \geq 2$, $B$ the event $X+Z \geq 3$. Note that $A \backslash B$ is $X=2,Z=0$ and has probability $6p^2(1-p)^4$. Similarly, $B \backslash A$ is $X=1,Z=2$ and has probability $4p^3(1-p)^3$.

Now, $P(A)=P(B)$ iff $P(A \backslash B)=P(B \backslash A)$ iff $6(1-p)=4p$ iff $p=3/5$.