Let E be the even the sum of dice is even and let F be the e

Let E be the even \"the sum of dice is even\" and let F be the even \"the sum of dice is greater than 8\" are E and F independent?

Solution

solution :

let say we roll two dice simultaneosly,

E be the even \"the sum of dice is even\"

and

F be the even \"the sum of dice is greater than 8\"

so E will be even when the two dice sum =any of {2,4,6,8,10,12}

and F will be even when any of {9,10,11,12} occurs.

for independency the two event must be disjoint. i.e the probability distribution of E given we know F should not depend on F / or the probability distribution of F given we know E should not depend on E.

or symobilcally ;

P(E|F) = P(E)

P(F|E) = P(F)

now if we look our experiment carefully ,

let say we have infromation about F, that is event F has occured , then of course from the set E u will not claim any number to happen wh[ich are less than 8, since F is already happened. based on this argument if F happens we modify our distribution set for E= {10 ,12}.this employs F affects E. so they are dependent.!