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A die marked $1,2,3$ in red and $4,5,6$ in green is tossed. Let $A$ be the event 'number is even' and $B$ be the event 'number is red'. Are $A$ and $B$ independent?

I know that $A\cap B={2}$ and $P(A) = \dfrac12$, $P(B) = \dfrac12$ and $A\cap B \neq P(A)\cdot P(B)$. So both these events are not independent. But I am having some doubts that if I go with the definition of independent events it says that 'Two events are independent if happening of one event does not affect happening of the other event'. In above case getting the number even does not affect getting a number which is red, so why are they are not independent.

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  • $\begingroup$ The mathematical definition is the correct one, that $P(A\cap B)=P(A)\cdot P(B)$ if and only if they are independent events. The intuitive approach can lead you to errors such as here. That being said, the intuitive approach can still lead you to the correct answer if you are careful about it... recognize that if you roll the die and you don't look at the result but someone else looks at it and tells you whether it is green or not, if you are told it is green you know it must be $4,5,$ or $6$...most of those are even, so being told color gives information as to chances of being even/odd. $\endgroup$ – JMoravitz Apr 10 '18 at 14:32
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    $\begingroup$ If you know the number is red, then prob of being even is only 1/3. Similarly if you know the number is even, then prob or being red is only 1/3. This shows they are dependent. Does this answer make intuitive sense to you? $\endgroup$ – antkam Apr 10 '18 at 14:32
  • $\begingroup$ Can we know which event is dependent or independent just by looking and reading rather than mathematical approach? $\endgroup$ – user190625 Apr 10 '18 at 14:42
  • $\begingroup$ @JMoravitz sorry I was not able to get it earlier but now I understand it thanks sir for clearing this out. $\endgroup$ – user190625 Apr 10 '18 at 15:19
  • $\begingroup$ @antkam thanks sir got it $\endgroup$ – user190625 Apr 10 '18 at 15:19
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In above case getting number even does not affect getting a number which is red,so why are they are not independent.

Actually, it does. If you don't know anything about the number that has just been thrown, the chance of the die being red is $1/2$ in your eyes. If you know the number is even, then the chances of the die being red decreases to just $1/3$, since only one of the three even numbers is on the red die. So knowing about one event gives you information about the other. Therefore they are not independent.

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  • $\begingroup$ Ok now I got it completely thanks sir $\endgroup$ – user190625 Apr 10 '18 at 15:17

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