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Let $\Omega = \{(i,j): i,j\in \{1,2,3,4,5,6\}~\}$, $A= \{(i,j):i\in\{1,3,5\}\},$ and $B=\{(i,j): i+j=9\}$.

My answer was : A and B are not independent because you can see clearly that B depends on A to get $i+j=8$. So B depends on A to get the values for A thus making A & B dependent.

How can I show they're not dependent by definition?

This is the information I know: $P(A)=\frac{1}{2}$, $P(B)=\frac{5}{36}$ and $P(A \cap B)= \frac{1}{18}$, and $P(A|B)= \frac{2}{5}$.

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    For general events $A,B$, what is the _definition_ of the statement "A and B are independent"? You may need to look it up in your textbook. – 2017-02-27
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    "My professor told me to use the definition, what does he mean by that?" Hmmm... maybe they mean that you should look up the definition and then use it? – 2017-02-27
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    @Killercamin with the new information you added (see answer below) it's immediate to verify that A and B are **not** independent, since $P(A)P(B)\neq P(A\cap B)$. – 2017-02-28
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    so as simple as that? I just state the the multiplication of both probabilities is not equal to the intersection? – 2017-02-28
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    @Killercamin Once again: **check your definitions**. – 2017-02-28
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    Out of curiosity, who told you that $P(B)=\frac5{36}$? Or did you evaluate it yourself? This could make all the difference... – 2017-02-28
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    Professor evaluated it – 2017-02-28
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    This does not answer my query, I am afraid. (But anyway, next time, please use @.) – 2017-02-28

2 Answers 2

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I think you should do some reading on independence of two events first, for example, Independence

So he wants you to check manually whether this equation holds for your A and B.

P(A∩B)=P(A)P(B) .

So, compute P(A) and P(B), then find the intersection and compute the probability of P(A∩B) and then check if the equality holds.

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A and B are independent if and only if $P(A\cap B)=P(A)P(B)$.

In general, one cannot say whether two events are independent or not without knowing the actual probability distribution (though there are some exceptions). In fact, in your case, there are two different probability distributions that one can associate to $\Omega$, such that A and B are independent for one, but not the other.

If $P(i,j)=1$ for $i=j=1$, and $P(i,j)=0$ otherwise, then $P(B)=0$ and $P(A\cap B)=P(A)P(B)=0$; thus $A$ and $B$ are independent.

If $P(i,j)=1/2$ for $(i,j)=(5,4)$ or $(i,j)=(6,4)$ and $P(i,j)=0$ otherwise, then $1/2=P(A)=P(B)=P(A\cap B)\neq P(A)P(B)=1/4$; thus $A$ and $B$ are not independent.

But if you know $P(A)=\frac{1}{2}$, $P(B)=\frac{5}{36}$ and $P(A\cap B)=\frac{1}{18}$, it's immediate to verify that $\frac{1}{2}\cdot\frac{5}{36}\neq \frac{1}{18}$ and thus A and B are not indepedent.