There is a game involving moving a game piece on a number line. The way to move the piece is determined by rolling a fair die, with the following rules: (I) When the die shows 1 point, the piece does not move. (II) When the die shows 3 or 5 points, the piece moves left (negative direction) by ``that point number minus 1'' units. (III) When the die shows an even number, the piece moves right (positive direction) by ``half of that point number'' units. On the first die roll, the piece starts at the origin. From the second roll onwards, the piece starts from the position it was in after the previous roll. For example, if two die rolls result in 5 points and 2 points respectively, the piece first moves left 4 units to coordinate $-4$, then moves right 1 unit to coordinate $-3$. Select the correct options. (1) After rolling the die once, the probability that the piece is at distance 2 from the origin is $\frac { 1 } { 2 }$ (2) After rolling the die once, the expected value of the piece's coordinate is 0 (3) After rolling the die twice, the piece's coordinate could be $-5$ (4) After rolling the die twice, given that the sum of the two rolls is odd, the probability that the piece's coordinate is positive is $\frac { 4 } { 9 }$ (5) After rolling the die three times, the probability that the piece is at the origin is $\left( \frac { 1 } { 6 } \right) ^ { 3 }$
There is a game involving moving a game piece on a number line. The way to move the piece is determined by rolling a fair die, with the following rules:\\
(I) When the die shows 1 point, the piece does not move.\\
(II) When the die shows 3 or 5 points, the piece moves left (negative direction) by ``that point number minus 1'' units.\\
(III) When the die shows an even number, the piece moves right (positive direction) by ``half of that point number'' units.\\
On the first die roll, the piece starts at the origin. From the second roll onwards, the piece starts from the position it was in after the previous roll. For example, if two die rolls result in 5 points and 2 points respectively, the piece first moves left 4 units to coordinate $-4$, then moves right 1 unit to coordinate $-3$. Select the correct options.\\
(1) After rolling the die once, the probability that the piece is at distance 2 from the origin is $\frac { 1 } { 2 }$\\
(2) After rolling the die once, the expected value of the piece's coordinate is 0\\
(3) After rolling the die twice, the piece's coordinate could be $-5$\\
(4) After rolling the die twice, given that the sum of the two rolls is odd, the probability that the piece's coordinate is positive is $\frac { 4 } { 9 }$\\
(5) After rolling the die three times, the probability that the piece is at the origin is $\left( \frac { 1 } { 6 } \right) ^ { 3 }$