# Why is that the events (Sum of dice roll=6, first die=4) are dependent, but the events (Sum of dice roll=7, first die =4) are independent?

Roll $2$ dice.
Let $E$ be the event that the sum of the dice is $6$
Let $F$ be the event that the sum of the dice is $7$
Let $G$ be the event that the first die rolled is a $4$

$E$ and $G$ are dependent (since $P(E\cap G) \neq P(E)P(G)$ )
$F$ and $G$ are independent (since $P(F\cap G) = P(F)P(G)$ )

Intuitively, why is this true?

• In neither cases are they causally independent, which I think relates to your intuition. But in the latter case they are uncorrelated. – Lucas Dec 10 '12 at 22:38
• @Lucas: You say that $F$ and $G$ are uncorrelated. What do you mean by that? I know that the correlation applies to random variables, but here it is about events. Also, $P(F\cap G) = P(F)P(G)$ means, by definition, that $F$ and $G$ are independent events. What do you mean that they are not "causally independent"? – digital-Ink Sep 17 '16 at 12:51
• @digital-Ink good question. In that comment I was language to highlight the distinction between causation and correlation - and avoid using the terms dependent independent. So I was using correlation loosely here (though you could assign numerical values to the events and speak of correlation properly) I understood it to be implicit in the question that the first intuition would be that neither are be dependent - because neither dice affects (causally) the other. The question of what causation is is complex. Here, by causally independent I mean something not physically affecting something else – Lucas Sep 17 '16 at 20:28

Roughly speaking, if you know the sum of the dice is $6$, that gives you some information about what the first die rolled: namely, that it couldn't possibly have rolled a $6$. On the other hand, if you know the sum of the dice is $7$, that doesn't tell you anything about the result of the first die. For any number $n$ it might have rolled, the second die could have rolled the number $7-n$. So events relating to the result of the first die are independent of whether you rolled a $7$, but not of whether you rolled a $6$.

You can make this a little more rigorous by thinking in terms of conditional probabilities: if $A$ and $B$ are events with nonzero probability, then they're independent if and only if $P(A|B)=P(A)$ (or $P(B|A)=P(B)$). For your events, a bit of case analysis will show you that $P(G|F)=1/6=P(G)$, but $P(G|E)=1/5 \neq P(G)$.

We want to understand why P(F | G) = P(F). There are exactly 6 pairs of dice values whose sum is 7: {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}. That is for each possible outcome of the first roll, there is exactly one (unique) outcome of the second roll which would yield a sum of 7. In other words, given the value of the first die, the probability that the sum of the dice roll is 7 is 1/6.

P(E|G) != P(E) because there are values of the first roll which make it impossible to have a sum of 6 - in particular, if the value of the first roll is 6. Therefore, knowledge of the value of the first die changes the probability that the sum of the two dice is 7.

Think about it this way: when you roll a any number the first time, like a 4, there is only 1 out of 6 possibilities that the next number will give you a sum of seven, thus 1/6

Like Micah said, there are exactly 6 combinations of the two die that will give you seven, out of a total of 36 possible combos. 6/36=1/6

Note the probability of getting a sum of seven with two die, is equal to the probability of getting a sum of seven after rolling 4. That is the definition of independence.

Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring

or P(B|A) = P(B)

Coming to our example,

𝐸 -> the sum of the dice is 6

F -> the sum of the dice is 7

G -> the first die rolled is a 4

F is a special case because we have 6 different ways of getting a sum of 7 unlike any other number.(Think of the sample space for any other sum.)

Lets look at the sample space of F

F = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}

Notice that, irrespective of what the number is on the first or second die, the probability of getting a sum of seven is a constant.

This isn't the case for event E (the event that the sum of the dice is 6), let's confirm the sample space

E = {(1,5), (2,4), (3,3), (4,2), (5,1)}

Notice that, the event of getting a 6 on either die excludes the possibility of getting a sum of 6 and hence our probability is influenced by/dependent on that.

This, IMHO, is the intuitive reason for the independence of events F and G.