Bounds on multinomial distribution

Question

Let $(X_1,X_2,X_3,Y)$ follow a multinomial distribution with parameters $n$ and $(p, p, p, 1-3p)$. Individually, each of $X_1, X_2, X_3$ follow a binomial distribution of with parameters $n$ and $p$ ; their median and expected value is $np$, but they are not independent (they are negatively associated).

1. What is $\mathrm{Pr}(X_1 \leq np \wedge X_2 \leq np \wedge X_3 \leq np)$ ?
2. What is $\mathrm{Pr}(X_1 + X_2 \leq 2np \wedge X_1+X_3 \leq 2np \wedge X_2+X_3 \leq 2np)$ ?

If the exact values are hard to obtain, good lower-bounds could be enough...

Edit: for 1. negative association enables to show that: $$ \mathrm{Pr}(X_1 \leq np \wedge X_2 \leq np \wedge X_3 \leq np) \leq \mathrm{Pr}(X_1 \leq np)^3 $$ and $\mathrm{Pr}(X_1 \leq np)$ should be close to $\frac{1}{2}$, but I need a lower-bound...