If `X and Y` are independent binomial vatiates `B(5,1/2) and B(7,1/2)` and the value of `P(X+Y=3)` is

Options

A. `(35)/(47)`
B. `(55)/(1024)`
C. `(220)/(512)`
D. `(11)/(204)`

Answer

B

Explanation

(b) We have,
`P(X=r)=""^(5)C_(r )((1)/(2))^(5)`
and `" " P(Y=r)=""^(7)C_(r )((1)/(2))^(7)`
Now, `" " P(X+Y=3)=P(X=0,Y=3)+P(X=1,Y=2)+P(X=2,Y=1)+P(X=3,Y=0)`
`=P(X=0)P(Y=3)+P(X=1)P(Y=2)+P(X=2)P(Y=1)+P(X=3)P(Y=0)`
`""^(5)C_(0)xx((1)/(2))^(5)xx""^(7)C_(3)xx((1)/(2))^(7)+""^(5)C_(1)xx((1)/(2))^(5)xx""^(7)C_(2)xx((1)/(2))^(7)+""^(5)C_(2)xx((1)/(2))^(5)xx""^(7)C_(1)xx((1)/(2))^(7)+""^(5)C_(3)xx((1)/(2))^(5)xx""^(7)C_(0)xx((1)/(2))^(7)`
`=(1)/(2^(12))(1xx35+5xx21+10xx7+10xx1)`
`=(220)/(2^(12))=(55)/(1024)`