Let $X_1,\ldots,X_{735},Y_1,\ldots,Y_{880}$ be independent random variables such that $P(X_i=0)=\frac{3}{7}$, $P(X_i=1)=\frac{4}{7}$ and $P(Y_i=0)=P(Y_i=1)=\frac{1}{2}$. Find $P(\sum_{i=1}^{735} X_i < \sum_{i=1}^{880} Y_i)$ using CLT approximation.
My problem with this exercise is that the number of $X_i$'s (735) is different than the number of $Y_i$'s (880). Could you please give a hint how to solve this problem? I would greatly appreciate any help...
Since $\frac{\sum X_i-np}{\sqrt{np(1-p)}}$~$N(0,1)$, so $\sum X_i$~$N(np,np(1-p))$.
Then you get $(\sum X_i-\sum Y_j)$~$N(np-mq,np(1-p)+mq(1-q))$