Question

Imagine rolling two fair 6 sided dice. the number rolled on the first die is even and the sum of the rolls is ten. are these two events independent?

Answer 1

even on first die : 2,4,6

P[ number rolled on the first die is even ] = 3/6 = 1/2

sum is 10 : (5,5), (6,4), (4,6) out of 36 possibilities

P[ sum of the rolls is ten ] = 3/36 = 1/12

sum is 10 and even on first die (6,4), (4,6)

P[ sum is 10 and even on first die ] = 2/36 = 1/18

For, the events being independent  

P[ sum is 10 and even on first die ] = P[ number rolled on the first die is even ]*P[ sum of the rolls is ten ]  

P[ number rolled on the first die is even ]*P[ sum of the rolls is ten ] = 1/2*1/12 = 1/24

which is not equal to P[ sum is 10 and even on first die ]

hence, the events are not independent