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Consider the event of rolling a die. The sample space is $\{1,2,3,4,5,6\}$.

Let $A$ be the event of rolling an odd number, i.e $\{1,3,5\}$.

Which of the following is not empty but is independent of $A$:

$ B = \{1,2\}, \ C = \{2,4,5\}, \ D = \{2,4,6\}. $

My answer was $D$, and in fact I think either one of the three answers should be correct. But the textbook says the answer is $B = \{1,2\}$. Could someone please explain?

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    $\begingroup$ What is the definition of independence? $\endgroup$
    – Glen_b
    Aug 21 '17 at 6:24
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I'm afraid you are missing independent events with incompatible events.

There are several equivalent definitions of independent events. A quite useful one is that two events are independent if knowing that one of them has happened doesn't give any information about the probability of the other.

Here, if A happens, B has not happened - that is, they are incompatible. Then, knowing that A has happened gives a lot of information about the probability of B: if you know the dice yielded 1, 3 or 5, you can be sure that it didn't yield 2, 4 or 6. Therefore, they aren't independent.

Using other common definitions of independence lead to the same conclusion that A and D are not independent (definitions given assuming events $A$ and $D$ are non empty with probability different from 0):

$P(A \cap D) = P(A)·P(D)$

This definition is not fulfilled because $P(A \cap D) = 0$ but $P(A)·P(D)=0.5·0.5=0.25$

Another common definition is that for independent events $P(A|D)=P(A)$ and $P(D|A)=P(D)$, and here $P(A|D)=0$ while $P(A)=0.5$ (and the same for D).

Please notice that the same reasoning can be used with any pair of incompatible events: incompatible events with probability different than zero are never independent.

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Think that if event $A$ occurs, then event $D$ cannot. (If an odd number is rolled, an even number cannot have been rolled).

Alternatively, if $A$ has already occurred, the probability of event $B$ occurring is equal to the probability of rolling a $1$ given that you rolled a $1, 3$ or $5$. i.e. $1 \over 3$. And if $A$ does not occur, then similarly the probability of event $B$ occurring is $1 \over 3$.

If you don't know anything about $A$ occurring or not, the probability of $B$ is still $1 \over 3$.

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