A property of sum of independent binomial random variables

Question

Define independent random variables $$X = B(n_1,p), Y = B(n_2,p), Z = B(n_1+n_2+1,p).$$ Given that $$Pr[X \geq a] > c, Pr[Y \geq b] > c,$$ where a and b are positive integers, and c is a constant in (0,1), can we show that $$Pr[Z \geq a+b] > c?$$ I've tried using convolution, but the resulting expression is too hard to simplify and produce the desired result.

Answer 1

The answer is no. If $n_1=n_2=2, p=\frac 12, a=b=2, c=0.24$ then

$$ \Pr(X\geqslant a)=\Pr(Y\geqslant b)=\frac 1 4>c. $$

But $$ \Pr(Z\geqslant a+b)=\frac{6}{32} <c. $$