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Two unbiased dice are rolled. Let A be the event that sum is $6$ and $B$ be the event that sum is $9$.

Are these events independent?

Intuitively it seems that these events are independent. But, $P(A)=5/36$ and $P(B) =4/36$ and $A \cap B = \phi$ so $P(A \cap B) =P(A) P(B) $ is not satisfied. What am i missing here?

Math Lover
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    Before reading the answers below, take your time to try articulating (to yourself or, even better, editing the question) what independence of events means; then, comparing your intuition against the 'correct' answers will be meaningful and instructive; I suggest this because this concept may be less straightforward than it cursorily seems. $\quad$ Also, take a look at points 1 & 3 of my explanation here: Common mistakes in probability. – ryang Mar 25 '23 at 09:27
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    Intuitively to me it seems that these events are not independent. And then you give an excellent reason for why they are not independent. – Henry Mar 25 '23 at 10:28
  • Does this answer your question? How can I intuitively see independent events? – ryang Mar 25 '23 at 16:58

3 Answers3

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The events $\{Sum=6\}$ and $\{Sum=9\}$ are disjoint (can't happen both in the same experiment), and information that one event occurred in the experiment, gives information that the other did not occur, so intuitively, the events are dependent.

I guess you confuse disjoint and independent.

kludg
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  • +1 See https://stats.stackexchange.com/a/380791/2958 for an illustration of the distinction – Henry Mar 25 '23 at 10:30
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Why does your intuition suggest that they're independent?

To put it more simply, if the sum of two dice is $6$, then can it ever be $9$?. So do you see the dependence now? If $A$ and $B$ were independent then $A$ and $B^{c}$ would also be independent. But here $A\subset B^{c}$ .

Mr.Gandalf Sauron
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A and B are dependent, because (as the question should be read) they are about the same dice roll. In fact they are exclusive events (the sum can't be both 6 and 9), which if the events can occur at all implies dependent (for the reason stated in the question).

What's independent is the outcome of distinct dice rolls. Like: the event that in a first throw of two dices they sum to 6 is independent of the event than in another throw they sum to 9.

fgrieu
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