We have sin a + cos a = 1/3

Use the fact that sin 2a = 2*(sin a)(cos a)

sin a + cos a = 1/3

square the right and left hand sides

(sin a + cos a)^2 = (1/3)^2

=> (sin a)^2 + (cos a)^2 + 2*(sin a)(cos a) = 1/9

(sin a)^2 + (cos a)^2 = 1

=> 1 + 2*(sin a)(cos a) = 1/9

=> 2*(sin a)(cos a) = 1/9 - 1

=> 2*(sin a)(cos a) = -8/9

=> sin 2a = -8/9

**The required value of sin 2a = -8/9**