We have two urns $U$ and $V$ containing balls. On each of the balls is written one of the numbers $-1$, $1$, or $2$.
Urn $U$ contains one ball bearing the number 1 and three balls bearing the number $-1$. Urn $V$ contains one ball bearing the number 1 and three balls bearing the number 2. We consider a game in which each round proceeds as follows: first we draw at random a ball from urn $U$, we note $x$ the number written on this ball and then we put it in urn $V$. In a second step, we draw at random a ball from urn $V$ and we note $y$ the number written on this ball. We consider the following events:

• $U _ { 1 }$: ``we draw a ball bearing the number 1 from urn $U$, that is $x = 1$'';
• $U _ { - 1 }$: ``we draw a ball bearing the number $-1$ from urn $U$, that is $x = -1$'';
• $V _ { 2 }$: ``we draw a ball bearing the number 2 from urn $V$, that is $y = 2$'';
• $V _ { 1 }$: ``we draw a ball bearing the number 1 from urn $V$, that is $y = 1$'';
• $V _ { - 1 }$: ``we draw a ball bearing the number $-1$ from urn $V$'', that is $y = -1$''.

1. Copy and complete the probability tree.
2. In this game, with each round we associate the complex number $z = x + \mathrm { i } y$.
   Calculate the probabilities of the following events. The answers will be justified. a. $A$: ``$z = -1 - \mathrm { i }$''; b. $B$: ``$z$ is a solution of the equation $t ^ { 2 } + 2 t + 5 = 0$''; c. $C$: ``In the complex plane with an orthonormal coordinate system $( \mathrm { O } ; \vec { u } , \vec { v } )$ the point $M$ with affixe $z$ belongs to the disk with center O and radius 2''.
3. During a round, we obtain the number 1 on each of the balls drawn. Show that the complex number $z$ associated with this round satisfies $z ^ { 2020 } = - 2 ^ { 1010 }$.

When rolling a die twice, let the outcomes be $m$ and $n$ in order. If the probability that $i ^ { m } \cdot ( - i ) ^ { n } = 1$ is $\frac { q } { p }$, find the value of $p + q$. (Here, $i = \sqrt { - 1 }$ and $p , q$ are coprime natural numbers.) [4 points]