Question

If $X$ and $Y$ are independent binomial variates $B(5,1/2)$ and $B(7,1/2)$, then find the value of $P(X+Y=3)$.

• A. $P(X+Y=3)=\frac{45}{1024}$
• B. $P(X+Y=3)=\frac{65}{1024}$
• C. $P(X+Y=3)=\frac{55}{1024}$
• D. $P(X+Y=3)=\frac{75}{1024}$

Answer 1

The correct option is C $P(X+Y=3)=\frac{55}{1024}$

For
$X=0$ and $Y=3$,
we get
$P={}^5{C}_0{\frac{1}{2}}^5.{}^7{C}_3{\frac{1}{2}}^7$
$=\frac{35}{4096}$ ...(i)
For
$X=1$ and $Y=2$,
we get
$P={}^5{C}_1{\frac{1}{2}}^5.{}^7{C}_2{\frac{1}{2}}^7$
$=\frac{105}{4096}$ ...(ii)
For
$X=2$ and $Y=1$,
we get
$P={}^5{C}_2{\frac{1}{2}}^5.{}^7{C}_1{\frac{1}{2}}^7$
$=\frac{70}{4096}$ ...(iii)
For
$X=3$ and $Y=0$,
we get
$P={}^5{C}_3{\frac{1}{2}}^5.{}^7{C}_0{\frac{1}{2}}^7$
$=\frac{10}{4096}$ ...(iv)
Adding i ii, iii and iv.
we get
$\frac{35+105+70+10}{4096}$
$=\frac{220}{4096}$
$=\frac{55}{1024}$