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A die is thrown until the number 3 appears:

  • $X$ is the event "In the first thrown I do not get the number 3"
  • $Y$ is the event "In the first four thrown I do not get the number 3"
  • $(X∩Y) = Y $

Why?

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    To not roll a 3 in the first 4 rolls you need to not roll a 3 on the first roll. – 2017-01-21
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    $X\cap Y$ is different from $Y$ if it is possible to satisfy $Y$ while not satisfying $X$. Can you see that this is not possible in this case? – 2017-01-21

2 Answers 2

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The notation is not very precise, but $X\cap Y$ is the event "both $X$ and $Y$ happen". This means that "I do not get number 3 on the first thrown" AND "I do not get number 3 on the first four thrown", which is equivalent to say "I do not get number 3 on the first four thrown", which is $Y$.

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    Whenever **Y->X** then **(X∩Y)=Y**? – 2017-01-21
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    Yes, you can write $Y\Rightarrow X$ as $Y\subseteq X$ in your notation – 2017-01-21
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Note that $X\cap Y=Y$ if and only if $Y\subset X$.