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) = 1/36So 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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