Pairwise/Mutual Independence of Events: Tossing Red/Blue Dice

[drawar]

Homework Statement

Toss a pair of fair dice, one in red and the other is blue. Define the events
A={Red dice showing 1 or 2 or 3}
B={Red dice showing 3 or 4 or 5}
C={The sum of points on the two dice equals 9}
Are the 3 events pairwise independent? mutually independent? Justify your answer.

Homework Equations

Events A and B are independent if and only if P(A intersection B) = P(A) x P(B)

The Attempt at a Solution

P(A)=P(B)=1/2
P(C)=4/36=1/9 (Since 9=3+6=4+5=5+4=6+3)

P(A intersection B) = P(Red *dice* showing 3) = 1/6 *should be die*
P(A intersection C) = 1/36 (Since the only possible combination that results in 9 is 3+6)
Likewise, P(B intersection C) = 3/36 = 1/12

It can be seen that P(A intersection B) ≠ P(A) x P(B) so A and B are not independent. The same conclusion can be made for A and C, B and C.
Therefore, A, B, C are neither pairwise independent nor mutually independent.

Above is my first try at the problem but I'm not sure if it's correct or not. Would appreciate it if someone would help me clarify this, thanks!

 

[HallsofIvy]

Homework Statement

Toss a pair of fair dice, one in red and the other is blue. Define the events
A={Red dice showing 1 or 2 or 3}
B={Red dice showing 3 or 4 or 5}
C={The sum of points on the two dice equals 9}
Are the 3 events pairwise independent? mutually independent? Justify your answer.

They are neither independent nor mutually exclusive because all three include "red die shows 3". By the way, "red dice" is grammatically incorrect because "dice" is the plural of "die".

Homework Equations

Events A and B are independent if and only if P(A intersection B) = P(A) x P(B)

The Attempt at a Solution

P(A)=P(B)=1/2
P(C)=4/36=1/9 (Since 9=3+6=4+5=5+4=6+3)

P(A intersection B) = P(Red dice showing 3) = 1/6
P(A intersection C) = 1/36 (Since the only possible combination that results in 9 is 3+6)
Likewise, P(B intersection C) = 3/36 = 1/12

It can be seen that P(A intersection B) ≠ P(A) x P(B) so A and B are not independent. The same conclusion can be made for A and C, B and C.
Therefore, A, B, C are neither pairwise independent nor mutually independent.

Above is my first try at the problem but I'm not sure if it's correct or not. Would appreciate it if someone would help me clarify this, thanks!

 

[drawar]

My bad, should have been
A={Red die showing 1 or 2 or 3}
B={Red die showing 3 or 4 or 5}

Thanks for spotting them.

Apart from this stupid grammatical error, is there anything wrong with my reasoning?

 

[Ray Vickson]

My bad, should have been
A={Red die showing 1 or 2 or 3}
B={Red die showing 3 or 4 or 5}

Thanks for spotting them.

Apart from this stupid grammatical error, is there anything wrong with my reasoning?

Your reasoning is fine.

RGV