Case of a subtle independence (Probability Theory)

The notion of dependent and independent events is a common one in probability theory and it’s applications.

Most of us got introduced to probability through a Dice, it used to be the go to example back in my school. So in this post I promise to use only one dice to discuss a problem concerning independent and dependent events. Let’s dive in!
 
Throw a Dice twice and record it’s outcome like this (1st,2nd).
Our sample space in this case would contain all the 36 possible outcomes:
[Sample space is the set of all possible outcomes]
(1,1) (1,2) … (1,6)
(2,1) (2,2) … (2,6)

(6,1) (6,2) … (6,6)
Now consider a bunch of events:
Event A: Getting a 4 on the first throw.
A = { (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) }
P(A) = 6/36 = 1/6
Event B: Sum of the two numbers is 6.
B = { (1,5) (2,4) (3,3) (4,2) (5,1) }
P(B) = 5/36
Event C: Sum of the two numbers is 7.

C = { (1,6) (2,5) (3,4) (4,3) (5,2) (6,1) }
P(C) = 6/36 = 1/6
Event D: Sum of the two numbers is 8.

D = { (2,6) (3,5) (4,4) (5,3) (6,2) }
P(D) = 5/36
For two events to be independent of each other, the following condition should be satisfied:
P(X ∩ Y) = P(X) . P(Y)
Let’s try to evaluate it for the above defined events:
P(A ∩ B) = 1/36  is not equal to P(A).P(B) = 5/(36*6)
P(A ∩ D) = 1/36  is not equal to P(A).P(D) = 5/(36*6)
P(A ∩ C) = 1/36  is equal to P(A).P(C) = 1/36
So we get that A and B are dependent on each other. Same is the case for A and D.
But A and C seem to be independent events! Why’s that?
Does this peculiarity represent some sort of intuitive idea or is it just numbers playing some trick?
Why is that getting a 4 on the first throw related to getting a sum 6 and 8 but not 7?
In fact of all the possible sums we can get by throwing a dice twice  (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12), event A is independent of only getting sum 7. You can check this for yourself by plugging the numbers in the above formula!
What’s more, it’s not just about getting a 4 on the first throw. Change event A to the case of getting any other number (1, 2, 3, 5, 6) on the first throw and the results will still hold true!
So what’s so special about getting a sum of 7 in this case?
Think about event A and event B for a minute. Should they be dependent on each other?
Well, if event A has occurred (i.e. we got a 4 on the first throw), event B could still happen if we get a 2 on the second throw, thereby making a sum of 6. But if we dint get a 4 on the first throw, can we be sure that event B can still happen?
Well, if we got any of these numbers (1,2,3 or 5) yes it can. But if instead we got a 6 on the first throw itself, then the sum of two throws will always be greater than 6 and hence event B can’t happen. So you see, it depends. It depends on what we get in the first throw for event B to happen or not.
You can develop similar reasoning for other events of getting different values of sums. Except 7.  It’s special!
Take a look at the event C again:
C = { (1,6) (2,5) (3,4) (4,3) (5,2) (6,1) }
For a first throw, whatever number we get from the dice we can still hope to get a sum of 7! So it really doesn’t matter what number we get in reality, because whatever it is, it is equally probable (1/6) to get a sum of 7 from there on.
 
Only number 7 has this property from the set of possible sum values and hence only for getting sum 7, you don’t care what number you get in the first throw, your chances will still be one in six!
Cool! Right? Just by looking, we couldn’t have guessed if getting a sum of 7 was any differently related to getting a 4 on the first throw than getting a sum of 6 or 8! But it is and there’s logic behind that! ^_^
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