Independence, part 2

Probability Level 3

A fair die is thrown two times independently. Let XX and YY be the random variable indicating the outcomes of these two throws. Define Z=X+Y,Z = X + Y, and define UU as the remainder when ZZ is divided by 66.

Which of the following statements are true?

  1. XX and ZZ are independent.
  2. XX and UU are independent.
  3. ZZ and UU are independent.

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