Question

Test each pair of events for independence
1) A and D
2) A and E
3) B and D
4) B and E
5) B and F
6) C and F
7) A and B
8) D and F

Answer 1

The pair of events that are independent are:

2.) A and E.
P(AΠE) = 0.28
P(A)*P(E) = 0.7*0.4 = 0.28
Hence, A and E are independent.

3.) B and D.
P(BΠD)= 0.03
P(B)*P(D)= 0.1*0.3 = 0.03
Hence, A and E are independent.

6.) C and F.
P(CΠF)= 0.06
P(C)*P(F)= 0.2*0.3 = 0.06
Hence, A and E are independent.

But, 1,4,5,7 and 8 do not comply to this condition hence they are not independent.

Hope i helped u:)

Answer 2

Two arbitrary events A and B are said to be independent if any one of the following three equivalent conditions hold:
1. P(A ∩ B) = P(A)P(B)
2. P(A|B) = P(A) - B has no effect on A
3. P(B|A) = P(B) - A has no effect on B

2 and 3 are calculated after calculating 1. So check condition 1. If it satisfies, then they are independent, otherwise they aren't.

1) A and D
P(A ∩ D) = 0.2
P(A)P(D) = 0.7 x 0.3 = 0.21
So A and D are not independent.

2) A and E
P(A ∩ E) = 0.28
P(A)P(E) = 0.7 x 0.4 = 0.28
So A and E are independent.

3) B and D
P(B ∩ D) = 0.03
P(B)P(D) = 0.1 x 0.3 = 0.03
So B and D are independent.

4) B and E
P(B ∩ E) = 0.05
P(B)P(E) = 0.1 x 0.4 = 0.04
So B and E are not independent.

5) B and F
P(B ∩ F) = 0.02
P(B)P(F) = 0.1 x 0.3 = 0.03
So B and F are not independent.

6) C and F
P(C ∩ F) = 0.06
P(C)P(F) = 0.2 x 0.3 = 0.06
So C and F are independent.

7) A and B
P(A ∩ B) = 0 (Since they are mutually exclusive)
P(A)P(B) = 0.7 x 0.1 = 0.07
So A and B are not independent.

8) D and F
P(D ∩ F) = 0 (Since they are mutually exclusive)
P(D)P(F) = 0.3 x 0.3 = 0.09
So D and F are not independent.