Independence between RV and union of RVs

Question

If random variables $X$ and $Y$ are independent and $X$ and $Z$ are independent, are $X$ and $Y \cup Z$ independent?

Answer 1

No.

suppose you are tossing two coins, a penny and a dime. $X$ is the event that the penny comes up $H$, $Y$ is the event that the dime comes up $H$, and $Z$ is the event that the coins match. Then the events that comprise $Y\cup Z$ are $$HH,TH,TT\implies P(Y\cup Z)=\frac 34$$ Where $TH$, for example, denotes the event "the penny comes up $T$ and the dime comes up $H$".

But $$P\left(X\cap (Y\cup Z)\right)=\frac 14\neq P(X)\times P(Y\cup Z)$$

Answer 2

No. The event $A_ i,_j$ that $i,j$ have the same birthdays is pairwise independent, so $A_1,_2$ and $A_2,_3$ are independent, as well as $A_1,_2$ and $A_1,_3$, but $A_1,_2, A_2,_3$ and $A_3,_1$ are clearly not independent. As person 1 having the same birthday as person 2 and person 3 having the same birthday as person 1 implies 2 and 3 have the same birthdays.