Consider independently rolling two fair dice, one red and the other green. Let A be the event that the red die shows dots, B be the event that the green die shows dots, and C be the event that the total number of dots showing on the two dice is . Are these events pairwise independent (i. e., are A and B independent events, are A and C independent, and are B and C independent)

The events A, B and C are pairwise independent but not mutually independent.

Step-by-step explanation:

The sample space when you roll two dice is given below.

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

From the above, Total Number of Sample Space=36

A is the event that the Red die shows 3 dots.

A=(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

n(A)=6

B is the event that the green die shows 4 dots.

B=(1,4) (2,4) (3,4) (4,4) (5,4) (6,4)

n(B)=6

C is the event that the total number of dots showing on the two dice is 7.

C= (1,6) (2,5) (3,4) (4,3) (5,2) (6,1)

n(C)=6

Definition

Events A, B and C are pairwise independent if the following holds.

P(A∩B)=P(A)P(B)

P(A∩C)=P(A)P(C)

P(B∩C)=P(B)P(C)

A∩B ={(3,4)}

P(A∩B)=

P(A)=

P(B)=

P(A)P(B)=

A∩C={(3,4)}

P(A∩C)=

P(A)=

P(C)=

P(A)P(B)=

B∩C={(3,4)}

P(B∩C)=

P(B)=

P(C)=

P(B)P(C)=

Since the conditions below are true

P(A∩B)=P(A)P(B)

P(A∩C)=P(A)P(C)

P(B∩C)=P(B)P(C)

The sets A, B and C are pairwise independent.

DEFINITION

The sets A, B and C are mutually independent if:

P(A∩B∩C) = P(A)P(B)P(C)

A∩B∩C = {(3,4)}

P(A∩B∩C) =

P(A)P(B)P(C) =

Since P(A∩B∩C) ≠ P(A)P(B)P(C), the events bare not mutually independent.

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step-by-step explanation: