Probability questions?

Question

1. If E and F are independent, and F and G are independent, does it follow that E and G are independent?

2. If E and F are independent, E and G are independent, does it follow that E is independent of (F^G)? (where ^ represents intersection). What about (FuG)? (where u represents union).

I'm stumped! If someone could help out I'd be very grateful =). Thanks!

Answer 1

M = { 1,2,3,4,5,6,7,8,9,a,b,c}

E = {1,2,3}

F = {3,4,5,6}

p(E) = 3/12 = 1/4

p(F) = 4/12 = 1/3

P(F) * P(E) = 1/4 *1/3 = 1/12

P(E n F) = 1/12

E and F are independend

G = {3,7,8,9 }

E and G are independend too

G n F = { 3}

p(G n F ) = 1/12

P[(G n F) n E] = 1/12

P(G n F) * P (E) = 1/12 * 1/4 = 1/48

so (G n F) and E are dependend

  

  

Answer 2

1 No

say

E=flip coin

F=roll die

G=flip 2nd coin if 1st coin(E) is a HEAD

EF ind

FG ind

EG not ind

2.

Id say yes E ind of both

Answer 3

1. The answer is no.... E and G can still be dependent

2. that ones stumps me too